:: NDIFF_5 semantic presentation

REAL is non empty non trivial V49() complex-membered ext-real-membered real-membered V167() non bounded_below non bounded_above V251() set
NAT is non empty non trivial epsilon-transitive epsilon-connected ordinal V49() cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V167() countable denumerable left_end bounded_below Element of K6(REAL)
K6(REAL) is non trivial V49() set
COMPLEX is non empty non trivial V49() complex-membered V167() set
K7(REAL,REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7(REAL,REAL)) is non trivial V49() set
omega is non empty non trivial epsilon-transitive epsilon-connected ordinal V49() cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V167() countable denumerable left_end bounded_below set
K6(omega) is non trivial V49() set
K6(NAT) is non trivial V49() set
K7(NAT,REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7(NAT,REAL)) is non trivial V49() set
K206() is non empty set
K7(K206(),K206()) is Relation-like set
K7(K7(K206(),K206()),K206()) is Relation-like set
K6(K7(K7(K206(),K206()),K206())) is set
K7(REAL,K206()) is non trivial Relation-like V49() set
K7(K7(REAL,K206()),K206()) is non trivial Relation-like V49() set
K6(K7(K7(REAL,K206()),K206())) is non trivial V49() set
K212() is RLSStruct
the carrier of K212() is set
K6( the carrier of K212()) is set
K216() is Element of K6( the carrier of K212())
K7(K216(),K216()) is Relation-like set
K7(K7(K216(),K216()),REAL) is Relation-like complex-valued ext-real-valued real-valued set
K6(K7(K7(K216(),K216()),REAL)) is set
RAT is non empty non trivial V49() complex-membered ext-real-membered real-membered rational-membered V167() set
INT is non empty non trivial V49() complex-membered ext-real-membered real-membered rational-membered integer-membered V167() set
K7(COMPLEX,COMPLEX) is non trivial Relation-like complex-valued V49() set
K6(K7(COMPLEX,COMPLEX)) is non trivial V49() set
K7(K7(COMPLEX,COMPLEX),COMPLEX) is non trivial Relation-like complex-valued V49() set
K6(K7(K7(COMPLEX,COMPLEX),COMPLEX)) is non trivial V49() set
K7(K7(REAL,REAL),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7(K7(REAL,REAL),REAL)) is non trivial V49() set
K7(RAT,RAT) is non trivial Relation-like RAT -valued complex-valued ext-real-valued real-valued V49() set
K6(K7(RAT,RAT)) is non trivial V49() set
K7(K7(RAT,RAT),RAT) is non trivial Relation-like RAT -valued complex-valued ext-real-valued real-valued V49() set
K6(K7(K7(RAT,RAT),RAT)) is non trivial V49() set
K7(INT,INT) is non trivial Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued V49() set
K6(K7(INT,INT)) is non trivial V49() set
K7(K7(INT,INT),INT) is non trivial Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued V49() set
K6(K7(K7(INT,INT),INT)) is non trivial V49() set
K7(NAT,NAT) is non trivial Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued V49() set
K7(K7(NAT,NAT),NAT) is non trivial Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued V49() set
K6(K7(K7(NAT,NAT),NAT)) is non trivial V49() set
1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
K7(1,1) is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued V49() countable set
K6(K7(1,1)) is V49() V53() countable set
K7(K7(1,1),1) is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued V49() countable set
K6(K7(K7(1,1),1)) is V49() V53() countable set
K7(K7(1,1),REAL) is Relation-like complex-valued ext-real-valued real-valued set
K6(K7(K7(1,1),REAL)) is set
2 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
K7(2,2) is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued V49() countable set
K7(K7(2,2),REAL) is Relation-like complex-valued ext-real-valued real-valued set
K6(K7(K7(2,2),REAL)) is set
TOP-REAL 2 is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like strict constituted-Functions constituted-FinSeqs RLTopStruct
the carrier of (TOP-REAL 2) is non empty set
{} is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional complex-valued ext-real-valued real-valued natural-valued V49() V50() V53() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V134() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V167() Cardinal-yielding countable bounded_below bounded_above real-bounded V251() set
ExtREAL is non empty ext-real-membered V251() set
K7(COMPLEX,REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7(COMPLEX,REAL)) is non trivial V49() set
K764() is non empty strict multMagma
the carrier of K764() is non empty set
<REAL,+> is non empty strict V135() V136() V137() V139() left-invertible right-invertible invertible left-cancelable right-cancelable V277() multMagma
K442() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
multMagma(# REAL,K442() #) is strict multMagma
K770() is non empty strict V137() V139() left-cancelable right-cancelable V277() SubStr of <REAL,+>
<NAT,+> is non empty strict V135() V137() V139() left-cancelable right-cancelable V277() uniquely-decomposable SubStr of K770()
<REAL,*> is non empty strict V135() V137() V139() multMagma
K444() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
multMagma(# REAL,K444() #) is strict multMagma
<NAT,*> is non empty strict V135() V137() V139() uniquely-decomposable SubStr of <REAL,*>
0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional complex-valued ext-real-valued real-valued natural-valued V49() V50() V53() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V132() Function-yielding V134() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V167() Cardinal-yielding countable bounded_below bounded_above real-bounded V251() Element of NAT
3 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
- 1 is non empty V11() real ext-real non positive negative Element of REAL
Seg 1 is non empty trivial V49() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= 1 ) } is set
{1} is non empty trivial V49() V53() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded set
Seg 2 is non empty V49() 2 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= 2 ) } is set
{1,2} is non empty V49() V53() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded set
<*> REAL is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined REAL -valued Function-like one-to-one constant functional complex-valued ext-real-valued real-valued natural-valued V49() V50() V53() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V134() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V167() Cardinal-yielding countable bounded_below bounded_above real-bounded V251() FinSequence of REAL
Sum (<*> REAL) is V11() real ext-real Element of REAL
G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of G is non empty non trivial set
K7(REAL, the carrier of G) is non trivial Relation-like V49() set
K6(K7(REAL, the carrier of G)) is non trivial V49() set
F is non empty Relation-like REAL -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(REAL, the carrier of G))
dom F is non empty complex-membered ext-real-membered real-membered Element of K6(REAL)
f is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
X is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
X + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
1 / (X + 1) is non empty V11() real ext-real positive non negative V160() Element of RAT
K39((X + 1)) is non empty V11() real ext-real positive non negative set
K37(1,K39((X + 1))) is non empty V11() real ext-real positive non negative set
X is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
m is V11() real ext-real set
m " is V11() real ext-real Element of REAL
i is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
y0 is V11() real ext-real set
i + y0 is V11() real ext-real Element of REAL
i0 is V11() real ext-real set
(m ") + i0 is V11() real ext-real Element of REAL
1 / (m ") is V11() real ext-real Element of REAL
K39((m ")) is V11() real ext-real set
K37(1,K39((m "))) is V11() real ext-real set
i + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
1 / (i + 1) is non empty V11() real ext-real positive non negative V160() Element of RAT
K39((i + 1)) is non empty V11() real ext-real positive non negative set
K37(1,K39((i + 1))) is non empty V11() real ext-real positive non negative set
r is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
s is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
r + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
s + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
1 / (s + 1) is non empty V11() real ext-real positive non negative V160() Element of RAT
K39((s + 1)) is non empty V11() real ext-real positive non negative set
K37(1,K39((s + 1))) is non empty V11() real ext-real positive non negative set
1 / (r + 1) is non empty V11() real ext-real positive non negative V160() Element of RAT
K39((r + 1)) is non empty V11() real ext-real positive non negative set
K37(1,K39((r + 1))) is non empty V11() real ext-real positive non negative set
X . s is V11() real ext-real Element of REAL
(X . s) - 0 is V11() real ext-real Element of REAL
K38(0) is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional complex-valued ext-real-valued real-valued natural-valued V49() V50() V53() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V134() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V167() Cardinal-yielding countable bounded_below bounded_above real-bounded V251() set
K36((X . s),K38(0)) is V11() real ext-real set
|.((X . s) - 0).| is V11() real ext-real Element of REAL
lim X is V11() real ext-real Element of REAL
m is non empty Relation-like non-empty NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K6(K7(NAT,REAL))
F /* m is non empty Relation-like NAT -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of G))
K7(NAT, the carrier of G) is non trivial Relation-like V49() set
K6(K7(NAT, the carrier of G)) is non trivial V49() set
m " is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(m ") (#) (F /* m) is non empty Relation-like NAT -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of G))
lim ((m ") (#) (F /* m)) is Element of the carrier of G
0. G is V68(G) Element of the carrier of G
the ZeroF of G is Element of the carrier of G
i is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
((m ") (#) (F /* m)) . i is Element of the carrier of G
(((m ") (#) (F /* m)) . i) - (0. G) is Element of the carrier of G
- (0. G) is Element of the carrier of G
(((m ") (#) (F /* m)) . i) + (- (0. G)) is Element of the carrier of G
the addF of G is Relation-like K7( the carrier of G, the carrier of G) -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of G, the carrier of G), the carrier of G))
K7( the carrier of G, the carrier of G) is Relation-like set
K7(K7( the carrier of G, the carrier of G), the carrier of G) is Relation-like set
K6(K7(K7( the carrier of G, the carrier of G), the carrier of G)) is set
K462( the carrier of G, the addF of G,(((m ") (#) (F /* m)) . i),(- (0. G))) is Element of the carrier of G
||.((((m ") (#) (F /* m)) . i) - (0. G)).|| is V11() real ext-real non negative Element of REAL
the normF of G is non empty Relation-like the carrier of G -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of G,REAL))
K7( the carrier of G,REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of G,REAL)) is non trivial V49() set
the normF of G . ((((m ") (#) (F /* m)) . i) - (0. G)) is V11() real ext-real Element of REAL
m . i is V11() real ext-real Element of REAL
F /. (m . i) is Element of the carrier of G
F . (m . i) is Element of the carrier of G
(m . i) " is V11() real ext-real Element of REAL
((m . i) ") * (F /. (m . i)) is Element of the carrier of G
the Mult of G is Relation-like K7(REAL, the carrier of G) -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of G), the carrier of G))
K7(K7(REAL, the carrier of G), the carrier of G) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of G), the carrier of G)) is non trivial V49() set
K458( the Mult of G,((m . i) "),(F /. (m . i))) is set
||.(((m . i) ") * (F /. (m . i))).|| is V11() real ext-real non negative Element of REAL
the normF of G . (((m . i) ") * (F /. (m . i))) is V11() real ext-real Element of REAL
abs ((m . i) ") is V11() real ext-real Element of REAL
||.(F /. (m . i)).|| is V11() real ext-real non negative Element of REAL
the normF of G . (F /. (m . i)) is V11() real ext-real Element of REAL
(abs ((m . i) ")) * ||.(F /. (m . i)).|| is V11() real ext-real Element of REAL
|.(m . i).| is V11() real ext-real Element of REAL
|.(m . i).| " is V11() real ext-real Element of REAL
(|.(m . i).| ") * ||.(F /. (m . i)).|| is V11() real ext-real Element of REAL
rng m is non empty complex-membered ext-real-membered real-membered Element of K6(REAL)
||.(((m ") (#) (F /* m)) . i).|| is V11() real ext-real non negative Element of REAL
the normF of G . (((m ") (#) (F /* m)) . i) is V11() real ext-real Element of REAL
(F /* m) . i is Element of the carrier of G
(m ") . i is V11() real ext-real Element of REAL
((m ") . i) * ((F /* m) . i) is Element of the carrier of G
K458( the Mult of G,((m ") . i),((F /* m) . i)) is set
||.(((m ") . i) * ((F /* m) . i)).|| is V11() real ext-real non negative Element of REAL
the normF of G . (((m ") . i) * ((F /* m) . i)) is V11() real ext-real Element of REAL
((m . i) ") * ((F /* m) . i) is Element of the carrier of G
K458( the Mult of G,((m . i) "),((F /* m) . i)) is set
||.(((m . i) ") * ((F /* m) . i)).|| is V11() real ext-real non negative Element of REAL
the normF of G . (((m . i) ") * ((F /* m) . i)) is V11() real ext-real Element of REAL
i0 is V11() real ext-real Element of REAL
f is non empty Relation-like non-empty NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K6(K7(NAT,REAL))
lim f is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional complex-valued ext-real-valued real-valued natural-valued V49() V50() V53() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V134() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V167() Cardinal-yielding countable bounded_below bounded_above real-bounded V251() Element of REAL
X is V11() real ext-real Element of REAL
m is V11() real ext-real Element of REAL
i is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
i0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
F /* f is non empty Relation-like NAT -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of G))
f " is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(f ") (#) (F /* f) is non empty Relation-like NAT -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of G))
rng f is non empty complex-membered ext-real-membered real-membered Element of K6(REAL)
y0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
((f ") (#) (F /* f)) . y0 is Element of the carrier of G
(((f ") (#) (F /* f)) . y0) - (0. G) is Element of the carrier of G
(((f ") (#) (F /* f)) . y0) + (- (0. G)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(((f ") (#) (F /* f)) . y0),(- (0. G))) is Element of the carrier of G
||.((((f ") (#) (F /* f)) . y0) - (0. G)).|| is V11() real ext-real non negative Element of REAL
the normF of G . ((((f ") (#) (F /* f)) . y0) - (0. G)) is V11() real ext-real Element of REAL
f . y0 is V11() real ext-real Element of REAL
(f . y0) - 0 is V11() real ext-real Element of REAL
K36((f . y0),K38(0)) is V11() real ext-real set
|.((f . y0) - 0).| is V11() real ext-real Element of REAL
|.(f . y0).| is V11() real ext-real Element of REAL
|.(f . y0).| " is V11() real ext-real Element of REAL
F /. (f . y0) is Element of the carrier of G
F . (f . y0) is Element of the carrier of G
||.(F /. (f . y0)).|| is V11() real ext-real non negative Element of REAL
the normF of G . (F /. (f . y0)) is V11() real ext-real Element of REAL
(|.(f . y0).| ") * ||.(F /. (f . y0)).|| is V11() real ext-real Element of REAL
(f . y0) " is V11() real ext-real Element of REAL
abs ((f . y0) ") is V11() real ext-real Element of REAL
(abs ((f . y0) ")) * ||.(F /. (f . y0)).|| is V11() real ext-real Element of REAL
((f . y0) ") * (F /. (f . y0)) is Element of the carrier of G
K458( the Mult of G,((f . y0) "),(F /. (f . y0))) is set
||.(((f . y0) ") * (F /. (f . y0))).|| is V11() real ext-real non negative Element of REAL
the normF of G . (((f . y0) ") * (F /. (f . y0))) is V11() real ext-real Element of REAL
(F /* f) . y0 is Element of the carrier of G
((f . y0) ") * ((F /* f) . y0) is Element of the carrier of G
K458( the Mult of G,((f . y0) "),((F /* f) . y0)) is set
||.(((f . y0) ") * ((F /* f) . y0)).|| is V11() real ext-real non negative Element of REAL
the normF of G . (((f . y0) ") * ((F /* f) . y0)) is V11() real ext-real Element of REAL
(f ") . y0 is V11() real ext-real Element of REAL
((f ") . y0) * ((F /* f) . y0) is Element of the carrier of G
K458( the Mult of G,((f ") . y0),((F /* f) . y0)) is set
||.(((f ") . y0) * ((F /* f) . y0)).|| is V11() real ext-real non negative Element of REAL
the normF of G . (((f ") . y0) * ((F /* f) . y0)) is V11() real ext-real Element of REAL
||.(((f ") (#) (F /* f)) . y0).|| is V11() real ext-real non negative Element of REAL
the normF of G . (((f ") (#) (F /* f)) . y0) is V11() real ext-real Element of REAL
lim ((f ") (#) (F /* f)) is Element of the carrier of G
f is V11() real ext-real Element of REAL
X is V11() real ext-real Element of REAL
G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of G is non empty non trivial set
K7(REAL, the carrier of G) is non trivial Relation-like V49() set
K6(K7(REAL, the carrier of G)) is non trivial V49() set
0. G is V68(G) Element of the carrier of G
the ZeroF of G is Element of the carrier of G
F is Relation-like REAL -defined the carrier of G -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of G))
F /. 0 is Element of the carrier of G
f is V11() real ext-real Element of REAL
X is V11() real ext-real Element of REAL
m is V11() real ext-real Element of REAL
|.m.| is V11() real ext-real Element of REAL
|.m.| " is V11() real ext-real Element of REAL
F /. m is Element of the carrier of G
||.(F /. m).|| is V11() real ext-real non negative Element of REAL
the normF of G is non empty Relation-like the carrier of G -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of G,REAL))
K7( the carrier of G,REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of G,REAL)) is non trivial V49() set
the normF of G . (F /. m) is V11() real ext-real Element of REAL
(|.m.| ") * ||.(F /. m).|| is V11() real ext-real Element of REAL
|.m.| * ((|.m.| ") * ||.(F /. m).||) is V11() real ext-real Element of REAL
|.m.| * f is V11() real ext-real Element of REAL
|.m.| * (|.m.| ") is V11() real ext-real Element of REAL
(|.m.| * (|.m.| ")) * ||.(F /. m).|| is V11() real ext-real Element of REAL
f * |.m.| is V11() real ext-real Element of REAL
1 * ||.(F /. m).|| is V11() real ext-real non negative Element of REAL
i is V11() real ext-real Element of REAL
i * |.m.| is V11() real ext-real Element of REAL
f * |.m.| is V11() real ext-real Element of REAL
F /. m is Element of the carrier of G
||.(F /. m).|| is V11() real ext-real non negative Element of REAL
the normF of G is non empty Relation-like the carrier of G -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of G,REAL))
K7( the carrier of G,REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of G,REAL)) is non trivial V49() set
the normF of G . (F /. m) is V11() real ext-real Element of REAL
m is V11() real ext-real Element of REAL
|.m.| is V11() real ext-real Element of REAL
F /. m is Element of the carrier of G
||.(F /. m).|| is V11() real ext-real non negative Element of REAL
the normF of G is non empty Relation-like the carrier of G -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of G,REAL))
K7( the carrier of G,REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of G,REAL)) is non trivial V49() set
the normF of G . (F /. m) is V11() real ext-real Element of REAL
f * |.m.| is V11() real ext-real Element of REAL
G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of G is non empty non trivial set
K7(REAL, the carrier of G) is non trivial Relation-like V49() set
K6(K7(REAL, the carrier of G)) is non trivial V49() set
F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of F is non empty non trivial set
K7( the carrier of G, the carrier of F) is Relation-like set
K6(K7( the carrier of G, the carrier of F)) is set
K7(REAL, the carrier of F) is non trivial Relation-like V49() set
K6(K7(REAL, the carrier of F)) is non trivial V49() set
f is Relation-like REAL -defined the carrier of G -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of G))
X is non empty Relation-like the carrier of G -defined the carrier of F -valued Function-like total quasi_total V153(G,F) V154(G,F) Lipschitzian Element of K6(K7( the carrier of G, the carrier of F))
X * f is Relation-like REAL -defined the carrier of F -valued Function-like Element of K6(K7(REAL, the carrier of F))
m is V11() real ext-real Element of REAL
dom X is non empty Element of K6( the carrier of G)
K6( the carrier of G) is set
rng f is Element of K6( the carrier of G)
dom f is complex-membered ext-real-membered real-membered Element of K6(REAL)
i is V11() real ext-real Element of REAL
i / 2 is V11() real ext-real Element of REAL
K39(2) is non empty V11() real ext-real positive non negative set
K37(i,K39(2)) is V11() real ext-real set
1 + m is V11() real ext-real Element of REAL
(i / 2) / (1 + m) is V11() real ext-real Element of REAL
K39((1 + m)) is V11() real ext-real set
K37((i / 2),K39((1 + m))) is V11() real ext-real set
y0 is V11() real ext-real Element of REAL
r is V11() real ext-real Element of REAL
|.r.| is V11() real ext-real Element of REAL
|.r.| " is V11() real ext-real Element of REAL
f /. r is Element of the carrier of G
||.(f /. r).|| is V11() real ext-real non negative Element of REAL
the normF of G is non empty Relation-like the carrier of G -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of G,REAL))
K7( the carrier of G,REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of G,REAL)) is non trivial V49() set
the normF of G . (f /. r) is V11() real ext-real Element of REAL
(|.r.| ") * ||.(f /. r).|| is V11() real ext-real Element of REAL
m + 1 is V11() real ext-real Element of REAL
(m + 1) * ((|.r.| ") * ||.(f /. r).||) is V11() real ext-real Element of REAL
(m + 1) * ((i / 2) / (1 + m)) is V11() real ext-real Element of REAL
y1 is V11() real ext-real Element of REAL
y1 + m is V11() real ext-real Element of REAL
s is V11() real ext-real Element of REAL
s + m is V11() real ext-real Element of REAL
m * ||.(f /. r).|| is V11() real ext-real Element of REAL
(m + 1) * ||.(f /. r).|| is V11() real ext-real Element of REAL
X . (f /. r) is Element of the carrier of F
||.(X . (f /. r)).|| is V11() real ext-real non negative Element of REAL
the normF of F is non empty Relation-like the carrier of F -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of F,REAL))
K7( the carrier of F,REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of F,REAL)) is non trivial V49() set
the normF of F . (X . (f /. r)) is V11() real ext-real Element of REAL
(|.r.| ") * ||.(X . (f /. r)).|| is V11() real ext-real Element of REAL
(|.r.| ") * ((m + 1) * ||.(f /. r).||) is V11() real ext-real Element of REAL
X /. (f /. r) is Element of the carrier of F
(X * f) /. r is Element of the carrier of F
||.((X * f) /. r).|| is V11() real ext-real non negative Element of REAL
the normF of F . ((X * f) /. r) is V11() real ext-real Element of REAL
(|.r.| ") * ||.((X * f) /. r).|| is V11() real ext-real Element of REAL
G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of G is non empty non trivial set
K7(REAL, the carrier of G) is non trivial Relation-like V49() set
K6(K7(REAL, the carrier of G)) is non trivial V49() set
0. G is V68(G) Element of the carrier of G
the ZeroF of G is Element of the carrier of G
F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of F is non empty non trivial set
K7( the carrier of G, the carrier of F) is Relation-like set
K6(K7( the carrier of G, the carrier of F)) is set
0. F is V68(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
K7(REAL, the carrier of F) is non trivial Relation-like V49() set
K6(K7(REAL, the carrier of F)) is non trivial V49() set
f is Relation-like REAL -defined the carrier of G -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of G))
f /. 0 is Element of the carrier of G
X is V11() real ext-real Element of REAL
m is Relation-like the carrier of G -defined the carrier of F -valued Function-like RestFunc-like Element of K6(K7( the carrier of G, the carrier of F))
m /. (0. G) is Element of the carrier of F
i is non empty Relation-like REAL -defined the carrier of G -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of G))
i + f is Relation-like REAL -defined the carrier of G -valued Function-like Element of K6(K7(REAL, the carrier of G))
m * (i + f) is Relation-like REAL -defined the carrier of F -valued Function-like Element of K6(K7(REAL, the carrier of F))
i0 is Element of the carrier of G
||.i0.|| is V11() real ext-real non negative Element of REAL
the normF of G is non empty Relation-like the carrier of G -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of G,REAL))
K7( the carrier of G,REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of G,REAL)) is non trivial V49() set
the normF of G . i0 is V11() real ext-real Element of REAL
dom m is Element of K6( the carrier of G)
K6( the carrier of G) is set
rng (i + f) is Element of K6( the carrier of G)
dom (i + f) is complex-membered ext-real-membered real-membered Element of K6(REAL)
dom (m * (i + f)) is complex-membered ext-real-membered real-membered Element of K6(REAL)
r is V11() real ext-real Element of REAL
r / 2 is V11() real ext-real Element of REAL
K39(2) is non empty V11() real ext-real positive non negative set
K37(r,K39(2)) is V11() real ext-real set
y0 is V11() real ext-real Element of REAL
1 + y0 is V11() real ext-real Element of REAL
(r / 2) / (1 + y0) is V11() real ext-real Element of REAL
K39((1 + y0)) is V11() real ext-real set
K37((r / 2),K39((1 + y0))) is V11() real ext-real set
y1 is V11() real ext-real Element of REAL
y1 / (1 + y0) is V11() real ext-real Element of REAL
K37(y1,K39((1 + y0))) is V11() real ext-real set
min (X,(y1 / (1 + y0))) is V11() real ext-real Element of REAL
a is V11() real ext-real Element of REAL
|.a.| is V11() real ext-real Element of REAL
f /. a is Element of the carrier of G
||.(f /. a).|| is V11() real ext-real non negative Element of REAL
the normF of G . (f /. a) is V11() real ext-real Element of REAL
1 * |.a.| is V11() real ext-real Element of REAL
i . a is Element of the carrier of G
a * i0 is Element of the carrier of G
the Mult of G is Relation-like K7(REAL, the carrier of G) -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of G), the carrier of G))
K7(K7(REAL, the carrier of G), the carrier of G) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of G), the carrier of G)) is non trivial V49() set
K458( the Mult of G,a,i0) is set
||.(i . a).|| is V11() real ext-real non negative Element of REAL
the normF of G . (i . a) is V11() real ext-real Element of REAL
y0 * |.a.| is V11() real ext-real Element of REAL
||.(i . a).|| - (y0 * |.a.|) is V11() real ext-real Element of REAL
K38((y0 * |.a.|)) is V11() real ext-real set
K36(||.(i . a).||,K38((y0 * |.a.|))) is V11() real ext-real set
(||.(i . a).|| - (y0 * |.a.|)) + (y0 * |.a.|) is V11() real ext-real Element of REAL
t is V11() real ext-real Element of REAL
t + (y0 * |.a.|) is V11() real ext-real Element of REAL
(i . a) + (f /. a) is Element of the carrier of G
the addF of G is Relation-like K7( the carrier of G, the carrier of G) -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of G, the carrier of G), the carrier of G))
K7( the carrier of G, the carrier of G) is Relation-like set
K7(K7( the carrier of G, the carrier of G), the carrier of G) is Relation-like set
K6(K7(K7( the carrier of G, the carrier of G), the carrier of G)) is set
K462( the carrier of G, the addF of G,(i . a),(f /. a)) is Element of the carrier of G
||.((i . a) + (f /. a)).|| is V11() real ext-real non negative Element of REAL
the normF of G . ((i . a) + (f /. a)) is V11() real ext-real Element of REAL
||.(i . a).|| + ||.(f /. a).|| is V11() real ext-real non negative Element of REAL
(y0 * |.a.|) + (1 * |.a.|) is V11() real ext-real Element of REAL
y0 + 1 is V11() real ext-real Element of REAL
(y0 + 1) * |.a.| is V11() real ext-real Element of REAL
((r / 2) / (1 + y0)) * ||.((i . a) + (f /. a)).|| is V11() real ext-real Element of REAL
((r / 2) / (1 + y0)) * ((y0 + 1) * |.a.|) is V11() real ext-real Element of REAL
(y0 + 1) * (y1 / (1 + y0)) is V11() real ext-real Element of REAL
m /. ((i . a) + (f /. a)) is Element of the carrier of F
||.(m /. ((i . a) + (f /. a))).|| is V11() real ext-real non negative Element of REAL
the normF of F is non empty Relation-like the carrier of F -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of F,REAL))
K7( the carrier of F,REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of F,REAL)) is non trivial V49() set
the normF of F . (m /. ((i . a) + (f /. a))) is V11() real ext-real Element of REAL
i /. a is Element of the carrier of G
(i /. a) + (f /. a) is Element of the carrier of G
K462( the carrier of G, the addF of G,(i /. a),(f /. a)) is Element of the carrier of G
m /. ((i /. a) + (f /. a)) is Element of the carrier of F
(i + f) /. a is Element of the carrier of G
m /. ((i + f) /. a) is Element of the carrier of F
(m * (i + f)) /. a is Element of the carrier of F
|.a.| " is V11() real ext-real Element of REAL
||.((m * (i + f)) /. a).|| is V11() real ext-real non negative Element of REAL
the normF of F . ((m * (i + f)) /. a) is V11() real ext-real Element of REAL
(|.a.| ") * ||.((m * (i + f)) /. a).|| is V11() real ext-real Element of REAL
((r / 2) / (1 + y0)) * (y0 + 1) is V11() real ext-real Element of REAL
(((r / 2) / (1 + y0)) * (y0 + 1)) * |.a.| is V11() real ext-real Element of REAL
(|.a.| ") * ((((r / 2) / (1 + y0)) * (y0 + 1)) * |.a.|) is V11() real ext-real Element of REAL
|.a.| * (|.a.| ") is V11() real ext-real Element of REAL
(|.a.| * (|.a.| ")) * ((r / 2) / (1 + y0)) is V11() real ext-real Element of REAL
((|.a.| * (|.a.| ")) * ((r / 2) / (1 + y0))) * (y0 + 1) is V11() real ext-real Element of REAL
1 * ((r / 2) / (1 + y0)) is V11() real ext-real Element of REAL
(1 * ((r / 2) / (1 + y0))) * (y0 + 1) is V11() real ext-real Element of REAL
G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of G is non empty non trivial set
K7(REAL, the carrier of G) is non trivial Relation-like V49() set
K6(K7(REAL, the carrier of G)) is non trivial V49() set
0. G is V68(G) Element of the carrier of G
the ZeroF of G is Element of the carrier of G
F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of F is non empty non trivial set
K7( the carrier of G, the carrier of F) is Relation-like set
K6(K7( the carrier of G, the carrier of F)) is set
0. F is V68(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
K7(REAL, the carrier of F) is non trivial Relation-like V49() set
K6(K7(REAL, the carrier of F)) is non trivial V49() set
f is Relation-like REAL -defined the carrier of G -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of G))
f /. 0 is Element of the carrier of G
X is Relation-like the carrier of G -defined the carrier of F -valued Function-like RestFunc-like Element of K6(K7( the carrier of G, the carrier of F))
X /. (0. G) is Element of the carrier of F
m is non empty Relation-like REAL -defined the carrier of G -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of G))
m + f is Relation-like REAL -defined the carrier of G -valued Function-like Element of K6(K7(REAL, the carrier of G))
X * (m + f) is Relation-like REAL -defined the carrier of F -valued Function-like Element of K6(K7(REAL, the carrier of F))
i is non empty Relation-like the carrier of G -defined the carrier of F -valued Function-like total quasi_total V153(G,F) V154(G,F) Lipschitzian Element of K6(K7( the carrier of G, the carrier of F))
i * f is Relation-like REAL -defined the carrier of F -valued Function-like Element of K6(K7(REAL, the carrier of F))
(i * f) + (X * (m + f)) is Relation-like REAL -defined the carrier of F -valued Function-like Element of K6(K7(REAL, the carrier of F))
G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of G is non empty non trivial set
K7(REAL, the carrier of G) is non trivial Relation-like V49() set
K6(K7(REAL, the carrier of G)) is non trivial V49() set
F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of F is non empty non trivial set
K7( the carrier of G, the carrier of F) is Relation-like set
K6(K7( the carrier of G, the carrier of F)) is set
f is V11() real ext-real Element of REAL
X is Relation-like REAL -defined the carrier of G -valued Function-like Element of K6(K7(REAL, the carrier of G))
X /. f is Element of the carrier of G
diff (X,f) is Element of the carrier of G
dom X is complex-membered ext-real-membered real-membered Element of K6(REAL)
m is complex-membered ext-real-membered real-membered open Neighbourhood of f
i is Relation-like the carrier of G -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of G, the carrier of F))
i * X is Relation-like REAL -defined the carrier of F -valued Function-like Element of K6(K7(REAL, the carrier of F))
K7(REAL, the carrier of F) is non trivial Relation-like V49() set
K6(K7(REAL, the carrier of F)) is non trivial V49() set
diff ((i * X),f) is Element of the carrier of F
diff (i,(X /. f)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F))
R_NormSpace_of_BoundedLinearOperators (G,F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators (G,F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators (G,F)))
R_VectorSpace_of_LinearOperators (G,F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators (G,F) is non empty functional Element of K6( the carrier of K266( the carrier of G,F))
K266( the carrier of G,F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of G, the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of G, the carrier of F
FuncZero ( the carrier of G,F) is Relation-like the carrier of G -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of G, the carrier of F)
0. F is V68(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
the carrier of G --> (0. F) is non empty Relation-like the carrier of G -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of G, the carrier of F))
FuncAdd ( the carrier of G,F) is Relation-like K7((Funcs ( the carrier of G, the carrier of F)),(Funcs ( the carrier of G, the carrier of F))) -defined Funcs ( the carrier of G, the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of G, the carrier of F)),(Funcs ( the carrier of G, the carrier of F))),(Funcs ( the carrier of G, the carrier of F))))
K7((Funcs ( the carrier of G, the carrier of F)),(Funcs ( the carrier of G, the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of G, the carrier of F)),(Funcs ( the carrier of G, the carrier of F))),(Funcs ( the carrier of G, the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of G, the carrier of F)),(Funcs ( the carrier of G, the carrier of F))),(Funcs ( the carrier of G, the carrier of F)))) is set
FuncExtMult ( the carrier of G,F) is Relation-like K7(REAL,(Funcs ( the carrier of G, the carrier of F))) -defined Funcs ( the carrier of G, the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of G, the carrier of F))),(Funcs ( the carrier of G, the carrier of F))))
K7(REAL,(Funcs ( the carrier of G, the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of G, the carrier of F))),(Funcs ( the carrier of G, the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of G, the carrier of F))),(Funcs ( the carrier of G, the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of G, the carrier of F)),(FuncZero ( the carrier of G,F)),(FuncAdd ( the carrier of G,F)),(FuncExtMult ( the carrier of G,F)) #) is strict RLSStruct
the carrier of K266( the carrier of G,F) is non empty set
K6( the carrier of K266( the carrier of G,F)) is set
K215(K266( the carrier of G,F),(LinearOperators (G,F))) is Relation-like Function-like Element of LinearOperators (G,F)
K213(K266( the carrier of G,F),(LinearOperators (G,F))) is Relation-like K7((LinearOperators (G,F)),(LinearOperators (G,F))) -defined LinearOperators (G,F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators (G,F)),(LinearOperators (G,F))),(LinearOperators (G,F))))
K7((LinearOperators (G,F)),(LinearOperators (G,F))) is Relation-like set
K7(K7((LinearOperators (G,F)),(LinearOperators (G,F))),(LinearOperators (G,F))) is Relation-like set
K6(K7(K7((LinearOperators (G,F)),(LinearOperators (G,F))),(LinearOperators (G,F)))) is set
K214(K266( the carrier of G,F),(LinearOperators (G,F))) is Relation-like K7(REAL,(LinearOperators (G,F))) -defined LinearOperators (G,F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators (G,F))),(LinearOperators (G,F))))
K7(REAL,(LinearOperators (G,F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators (G,F))),(LinearOperators (G,F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators (G,F))),(LinearOperators (G,F)))) is non trivial V49() set
RLSStruct(# (LinearOperators (G,F)),K215(K266( the carrier of G,F),(LinearOperators (G,F))),K213(K266( the carrier of G,F),(LinearOperators (G,F))),K214(K266( the carrier of G,F),(LinearOperators (G,F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators (G,F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators (G,F))) is set
K215((R_VectorSpace_of_LinearOperators (G,F)),(BoundedLinearOperators (G,F))) is Element of BoundedLinearOperators (G,F)
K213((R_VectorSpace_of_LinearOperators (G,F)),(BoundedLinearOperators (G,F))) is Relation-like K7((BoundedLinearOperators (G,F)),(BoundedLinearOperators (G,F))) -defined BoundedLinearOperators (G,F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators (G,F)),(BoundedLinearOperators (G,F))),(BoundedLinearOperators (G,F))))
K7((BoundedLinearOperators (G,F)),(BoundedLinearOperators (G,F))) is Relation-like set
K7(K7((BoundedLinearOperators (G,F)),(BoundedLinearOperators (G,F))),(BoundedLinearOperators (G,F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators (G,F)),(BoundedLinearOperators (G,F))),(BoundedLinearOperators (G,F)))) is set
K214((R_VectorSpace_of_LinearOperators (G,F)),(BoundedLinearOperators (G,F))) is Relation-like K7(REAL,(BoundedLinearOperators (G,F))) -defined BoundedLinearOperators (G,F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators (G,F))),(BoundedLinearOperators (G,F))))
K7(REAL,(BoundedLinearOperators (G,F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators (G,F))),(BoundedLinearOperators (G,F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators (G,F))),(BoundedLinearOperators (G,F)))) is non trivial V49() set
BoundedLinearOperatorsNorm (G,F) is non empty Relation-like BoundedLinearOperators (G,F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators (G,F)),REAL))
K7((BoundedLinearOperators (G,F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators (G,F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators (G,F)),K215((R_VectorSpace_of_LinearOperators (G,F)),(BoundedLinearOperators (G,F))),K213((R_VectorSpace_of_LinearOperators (G,F)),(BoundedLinearOperators (G,F))),K214((R_VectorSpace_of_LinearOperators (G,F)),(BoundedLinearOperators (G,F))),(BoundedLinearOperatorsNorm (G,F)) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F)) is non empty set
(diff (i,(X /. f))) . (diff (X,f)) is Element of the carrier of F
dom i is Element of K6( the carrier of G)
K6( the carrier of G) is set
0. G is V68(G) Element of the carrier of G
the ZeroF of G is Element of the carrier of G
i /. (X /. f) is Element of the carrier of F
i0 is Neighbourhood of X /. f
y0 is Relation-like the carrier of G -defined the carrier of F -valued Function-like RestFunc-like Element of K6(K7( the carrier of G, the carrier of F))
y0 /. (0. G) is Element of the carrier of F
y0 is Relation-like the carrier of G -defined the carrier of F -valued Function-like RestFunc-like Element of K6(K7( the carrier of G, the carrier of F))
y0 /. (0. G) is Element of the carrier of F
s is non empty Relation-like REAL -defined the carrier of G -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of G))
s . 1 is Element of the carrier of G
y1 is Relation-like REAL -defined the carrier of G -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of G))
s is non empty Relation-like REAL -defined the carrier of G -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of G))
s . 1 is Element of the carrier of G
y1 is Relation-like REAL -defined the carrier of G -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of G))
PP is Element of the carrier of G
(X /. f) - (X /. f) is Element of the carrier of G
- (X /. f) is Element of the carrier of G
(X /. f) + (- (X /. f)) is Element of the carrier of G
the addF of G is Relation-like K7( the carrier of G, the carrier of G) -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of G, the carrier of G), the carrier of G))
K7( the carrier of G, the carrier of G) is Relation-like set
K7(K7( the carrier of G, the carrier of G), the carrier of G) is Relation-like set
K6(K7(K7( the carrier of G, the carrier of G), the carrier of G)) is set
K462( the carrier of G, the addF of G,(X /. f),(- (X /. f))) is Element of the carrier of G
f - f is V11() real ext-real Element of REAL
K38(f) is V11() real ext-real set
K36(f,K38(f)) is V11() real ext-real set
s . (f - f) is Element of the carrier of G
y1 /. (f - f) is Element of the carrier of G
(s . (f - f)) + (y1 /. (f - f)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(s . (f - f)),(y1 /. (f - f))) is Element of the carrier of G
s . 0 is Element of the carrier of G
y1 /. 0 is Element of the carrier of G
(s . 0) + (y1 /. 0) is Element of the carrier of G
K462( the carrier of G, the addF of G,(s . 0),(y1 /. 0)) is Element of the carrier of G
DD is V11() real ext-real Element of REAL
DD * PP is Element of the carrier of G
the Mult of G is Relation-like K7(REAL, the carrier of G) -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of G), the carrier of G))
K7(K7(REAL, the carrier of G), the carrier of G) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of G), the carrier of G)) is non trivial V49() set
K458( the Mult of G,DD,PP) is set
(DD * PP) + (y1 /. 0) is Element of the carrier of G
K462( the carrier of G, the addF of G,(DD * PP),(y1 /. 0)) is Element of the carrier of G
(0. G) + (y1 /. 0) is Element of the carrier of G
K462( the carrier of G, the addF of G,(0. G),(y1 /. 0)) is Element of the carrier of G
r is non empty Relation-like the carrier of G -defined the carrier of F -valued Function-like total quasi_total V153(G,F) V154(G,F) Lipschitzian Element of K6(K7( the carrier of G, the carrier of F))
r * y1 is Relation-like REAL -defined the carrier of F -valued Function-like Element of K6(K7(REAL, the carrier of F))
s + y1 is Relation-like REAL -defined the carrier of G -valued Function-like Element of K6(K7(REAL, the carrier of G))
y0 * (s + y1) is Relation-like REAL -defined the carrier of F -valued Function-like Element of K6(K7(REAL, the carrier of F))
(r * y1) + (y0 * (s + y1)) is Relation-like REAL -defined the carrier of F -valued Function-like Element of K6(K7(REAL, the carrier of F))
r * s is non empty Relation-like REAL -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7(REAL, the carrier of F))
dom (r * s) is non empty complex-membered ext-real-membered real-membered Element of K6(REAL)
r . PP is Element of the carrier of F
tm is V11() real ext-real Element of REAL
s . tm is Element of the carrier of G
r . (s . tm) is Element of the carrier of F
tm * PP is Element of the carrier of G
K458( the Mult of G,tm,PP) is set
r . (tm * PP) is Element of the carrier of F
t is Element of the carrier of F
tm * t is Element of the carrier of F
the Mult of F is Relation-like K7(REAL, the carrier of F) -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of F), the carrier of F))
K7(K7(REAL, the carrier of F), the carrier of F) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of F), the carrier of F)) is non trivial V49() set
K458( the Mult of F,tm,t) is set
(r * s) . tm is Element of the carrier of F
Sw1 is complex-membered ext-real-membered real-membered open Neighbourhood of f
X .: Sw1 is Element of K6( the carrier of G)
w2 is complex-membered ext-real-membered real-membered open Neighbourhood of f
ys is set
rm is V11() real ext-real Element of REAL
X . rm is set
dom (i * X) is complex-membered ext-real-membered real-membered Element of K6(REAL)
ys is V11() real ext-real Element of REAL
X /. ys is Element of the carrier of G
(X /. ys) - (X /. f) is Element of the carrier of G
(X /. ys) + (- (X /. f)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(X /. ys),(- (X /. f))) is Element of the carrier of G
ys - f is V11() real ext-real Element of REAL
K36(ys,K38(f)) is V11() real ext-real set
s . (ys - f) is Element of the carrier of G
y1 /. (ys - f) is Element of the carrier of G
(s . (ys - f)) + (y1 /. (ys - f)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(s . (ys - f)),(y1 /. (ys - f))) is Element of the carrier of G
X . ys is set
dom y1 is complex-membered ext-real-membered real-membered Element of K6(REAL)
dom r is non empty Element of K6( the carrier of G)
rng y1 is Element of K6( the carrier of G)
dom (r * y1) is complex-membered ext-real-membered real-membered Element of K6(REAL)
dom (s + y1) is complex-membered ext-real-membered real-membered Element of K6(REAL)
dom y0 is Element of K6( the carrier of G)
rng (s + y1) is Element of K6( the carrier of G)
dom (y0 * (s + y1)) is complex-membered ext-real-membered real-membered Element of K6(REAL)
dom ((r * y1) + (y0 * (s + y1))) is complex-membered ext-real-membered real-membered Element of K6(REAL)
REAL /\ REAL is complex-membered ext-real-membered real-membered V251() set
r . (y1 /. (ys - f)) is Element of the carrier of F
r /. (y1 /. (ys - f)) is Element of the carrier of F
(r * y1) /. (ys - f) is Element of the carrier of F
y0 /. ((s . (ys - f)) + (y1 /. (ys - f))) is Element of the carrier of F
s /. (ys - f) is Element of the carrier of G
(s /. (ys - f)) + (y1 /. (ys - f)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(s /. (ys - f)),(y1 /. (ys - f))) is Element of the carrier of G
y0 /. ((s /. (ys - f)) + (y1 /. (ys - f))) is Element of the carrier of F
(s + y1) /. (ys - f) is Element of the carrier of G
y0 /. ((s + y1) /. (ys - f)) is Element of the carrier of F
(y0 * (s + y1)) /. (ys - f) is Element of the carrier of F
(r * s) . (ys - f) is Element of the carrier of F
r . (s . (ys - f)) is Element of the carrier of F
(i * X) /. ys is Element of the carrier of F
(i * X) /. f is Element of the carrier of F
((i * X) /. ys) - ((i * X) /. f) is Element of the carrier of F
- ((i * X) /. f) is Element of the carrier of F
((i * X) /. ys) + (- ((i * X) /. f)) is Element of the carrier of F
the addF of F is Relation-like K7( the carrier of F, the carrier of F) -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of F, the carrier of F), the carrier of F))
K7( the carrier of F, the carrier of F) is Relation-like set
K7(K7( the carrier of F, the carrier of F), the carrier of F) is Relation-like set
K6(K7(K7( the carrier of F, the carrier of F), the carrier of F)) is set
K462( the carrier of F, the addF of F,((i * X) /. ys),(- ((i * X) /. f))) is Element of the carrier of F
i /. (X /. ys) is Element of the carrier of F
(i /. (X /. ys)) - ((i * X) /. f) is Element of the carrier of F
(i /. (X /. ys)) + (- ((i * X) /. f)) is Element of the carrier of F
K462( the carrier of F, the addF of F,(i /. (X /. ys)),(- ((i * X) /. f))) is Element of the carrier of F
(i /. (X /. ys)) - (i /. (X /. f)) is Element of the carrier of F
- (i /. (X /. f)) is Element of the carrier of F
(i /. (X /. ys)) + (- (i /. (X /. f))) is Element of the carrier of F
K462( the carrier of F, the addF of F,(i /. (X /. ys)),(- (i /. (X /. f)))) is Element of the carrier of F
(diff (i,(X /. f))) . ((X /. ys) - (X /. f)) is Element of the carrier of F
y0 /. ((X /. ys) - (X /. f)) is Element of the carrier of F
((diff (i,(X /. f))) . ((X /. ys) - (X /. f))) + (y0 /. ((X /. ys) - (X /. f))) is Element of the carrier of F
K462( the carrier of F, the addF of F,((diff (i,(X /. f))) . ((X /. ys) - (X /. f))),(y0 /. ((X /. ys) - (X /. f)))) is Element of the carrier of F
(r . (s . (ys - f))) + (r . (y1 /. (ys - f))) is Element of the carrier of F
K462( the carrier of F, the addF of F,(r . (s . (ys - f))),(r . (y1 /. (ys - f)))) is Element of the carrier of F
((r . (s . (ys - f))) + (r . (y1 /. (ys - f)))) + ((y0 * (s + y1)) /. (ys - f)) is Element of the carrier of F
K462( the carrier of F, the addF of F,((r . (s . (ys - f))) + (r . (y1 /. (ys - f)))),((y0 * (s + y1)) /. (ys - f))) is Element of the carrier of F
((r * y1) /. (ys - f)) + ((y0 * (s + y1)) /. (ys - f)) is Element of the carrier of F
K462( the carrier of F, the addF of F,((r * y1) /. (ys - f)),((y0 * (s + y1)) /. (ys - f))) is Element of the carrier of F
(r . (s . (ys - f))) + (((r * y1) /. (ys - f)) + ((y0 * (s + y1)) /. (ys - f))) is Element of the carrier of F
K462( the carrier of F, the addF of F,(r . (s . (ys - f))),(((r * y1) /. (ys - f)) + ((y0 * (s + y1)) /. (ys - f)))) is Element of the carrier of F
tm is non empty Relation-like REAL -defined the carrier of F -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of F))
tm . (ys - f) is Element of the carrier of F
a is Relation-like REAL -defined the carrier of F -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of F))
a /. (ys - f) is Element of the carrier of F
(tm . (ys - f)) + (a /. (ys - f)) is Element of the carrier of F
K462( the carrier of F, the addF of F,(tm . (ys - f)),(a /. (ys - f))) is Element of the carrier of F
(r * s) . 1 is Element of the carrier of F
G is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of G is non empty set
F is Relation-like NAT -defined the carrier of G -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of G
len F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
dom F is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
Sum F is Element of the carrier of G
||.(Sum F).|| is V11() real ext-real non negative Element of REAL
the normF of G is non empty Relation-like the carrier of G -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of G,REAL))
K7( the carrier of G,REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of G,REAL)) is non trivial V49() set
the normF of G . (Sum F) is V11() real ext-real Element of REAL
f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
len f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
Sum f is V11() real ext-real Element of REAL
X is Relation-like NAT -defined the carrier of G -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of G
len X is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
dom X is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
Sum X is Element of the carrier of G
||.(Sum X).|| is V11() real ext-real non negative Element of REAL
the normF of G . (Sum X) is V11() real ext-real Element of REAL
m is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
len m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
Sum m is V11() real ext-real Element of REAL
K7(NAT, the carrier of G) is non trivial Relation-like V49() set
K6(K7(NAT, the carrier of G)) is non trivial V49() set
0. G is V68(G) Element of the carrier of G
the ZeroF of G is Element of the carrier of G
i is non empty Relation-like NAT -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of G))
i . (len X) is Element of the carrier of G
i . 0 is Element of the carrier of G
X is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
m is Relation-like NAT -defined the carrier of G -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of G
m | X is Relation-like NAT -defined the carrier of G -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of G
Seg X is V49() X -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= X ) } is set
m | (Seg X) is Relation-like NAT -defined Seg X -defined NAT -defined the carrier of G -valued Function-like V49() FinSubsequence-like countable set
i is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
i | X is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
i | (Seg X) is Relation-like NAT -defined Seg X -defined NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSubsequence-like countable set
X + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
len m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
len i is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
dom m is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
dom (m | X) is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
r is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
(i | X) . r is V11() real ext-real Element of REAL
(m | X) /. r is Element of the carrier of G
||.((m | X) /. r).|| is V11() real ext-real non negative Element of REAL
the normF of G . ((m | X) /. r) is V11() real ext-real Element of REAL
i . r is V11() real ext-real Element of REAL
m /. r is Element of the carrier of G
||.(m /. r).|| is V11() real ext-real non negative Element of REAL
the normF of G . (m /. r) is V11() real ext-real Element of REAL
m . r is set
(m | X) . r is set
Seg (X + 1) is non empty V49() X + 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= X + 1 ) } is set
i . (X + 1) is V11() real ext-real Element of REAL
m /. (X + 1) is Element of the carrier of G
||.(m /. (X + 1)).|| is V11() real ext-real non negative Element of REAL
the normF of G . (m /. (X + 1)) is V11() real ext-real Element of REAL
i /. (X + 1) is V11() real ext-real Element of REAL
<*(i /. (X + 1))*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing V49() 1 -element FinSequence-like FinSubsequence-like countable FinSequence of REAL
[1,(i /. (X + 1))] is set
{1,(i /. (X + 1))} is non empty V49() complex-membered ext-real-membered real-membered countable left_end right_end bounded_below bounded_above real-bounded set
{{1,(i /. (X + 1))},{1}} is non empty V49() V53() countable set
{[1,(i /. (X + 1))]} is non empty trivial Relation-like Function-like constant V49() 1 -element countable set
(i | X) ^ <*(i /. (X + 1))*> is non empty Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable Element of REAL *
REAL * is non empty functional FinSequence-membered FinSequenceSet of REAL
<*(i . (X + 1))*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing V49() 1 -element FinSequence-like FinSubsequence-like countable FinSequence of REAL
[1,(i . (X + 1))] is set
{1,(i . (X + 1))} is non empty V49() complex-membered ext-real-membered real-membered countable left_end right_end bounded_below bounded_above real-bounded set
{{1,(i . (X + 1))},{1}} is non empty V49() V53() countable set
{[1,(i . (X + 1))]} is non empty trivial Relation-like Function-like constant V49() 1 -element countable set
(i | X) ^ <*(i . (X + 1))*> is non empty Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable Element of REAL *
Sum i is V11() real ext-real Element of REAL
Sum (i | X) is V11() real ext-real Element of REAL
(Sum (i | X)) + (i . (X + 1)) is V11() real ext-real Element of REAL
m . (len m) is set
r is Element of the carrier of G
len (m | X) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
m | (dom (m | X)) is Relation-like NAT -defined dom (m | X) -defined NAT -defined the carrier of G -valued Function-like V49() FinSubsequence-like countable Element of K6(K7(NAT, the carrier of G))
K7(NAT, the carrier of G) is non trivial Relation-like V49() set
K6(K7(NAT, the carrier of G)) is non trivial V49() set
Sum m is Element of the carrier of G
Sum (m | X) is Element of the carrier of G
(Sum (m | X)) + r is Element of the carrier of G
the addF of G is Relation-like K7( the carrier of G, the carrier of G) -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of G, the carrier of G), the carrier of G))
K7( the carrier of G, the carrier of G) is Relation-like set
K7(K7( the carrier of G, the carrier of G), the carrier of G) is Relation-like set
K6(K7(K7( the carrier of G, the carrier of G), the carrier of G)) is set
K462( the carrier of G, the addF of G,(Sum (m | X)),r) is Element of the carrier of G
||.((Sum (m | X)) + r).|| is V11() real ext-real non negative Element of REAL
the normF of G . ((Sum (m | X)) + r) is V11() real ext-real Element of REAL
||.(Sum (m | X)).|| is V11() real ext-real non negative Element of REAL
the normF of G . (Sum (m | X)) is V11() real ext-real Element of REAL
||.r.|| is V11() real ext-real non negative Element of REAL
the normF of G . r is V11() real ext-real Element of REAL
||.(Sum (m | X)).|| + ||.r.|| is V11() real ext-real non negative Element of REAL
len (i | X) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
||.(Sum m).|| is V11() real ext-real non negative Element of REAL
the normF of G . (Sum m) is V11() real ext-real Element of REAL
m is Relation-like NAT -defined the carrier of G -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of G
len m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
i is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
len i is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
dom m is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
Sum m is Element of the carrier of G
||.(Sum m).|| is V11() real ext-real non negative Element of REAL
the normF of G . (Sum m) is V11() real ext-real Element of REAL
Sum i is V11() real ext-real Element of REAL
G is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of G is non empty set
F is Element of the carrier of G
f is Neighbourhood of F
X is Neighbourhood of F
f /\ X is Element of K6( the carrier of G)
K6( the carrier of G) is set
m is Neighbourhood of F
i is V11() real ext-real Element of REAL
{ b₁ where b₁ is Element of the carrier of G : not i <= ||.(b₁ - F).|| } is set
G is Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product G is non empty functional with_common_domain product-like set
F is set
dom G is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
f is Relation-like Function-like set
dom f is set
len G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
Seg (len G) is V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= len G ) } is set
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding set
product G is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like NORMSTR
the carrier of (product G) is non empty set
F is Element of the carrier of (product G)
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
zeros G is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
addop G is Relation-like Function-like BinOps of carr G
[:(addop G):] is Relation-like K7((product (carr G)),(product (carr G))) -defined product (carr G) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((product (carr G)),(product (carr G))),(product (carr G))))
K7((product (carr G)),(product (carr G))) is Relation-like set
K7(K7((product (carr G)),(product (carr G))),(product (carr G))) is Relation-like set
K6(K7(K7((product (carr G)),(product (carr G))),(product (carr G)))) is set
multop G is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr G
[:(multop G):] is Relation-like K7(REAL,(product (carr G))) -defined product (carr G) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(product (carr G))),(product (carr G))))
K7(REAL,(product (carr G))) is non trivial Relation-like V49() set
K7(K7(REAL,(product (carr G))),(product (carr G))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(product (carr G))),(product (carr G)))) is non trivial V49() set
productnorm G is non empty Relation-like product (carr G) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((product (carr G)),REAL))
K7((product (carr G)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((product (carr G)),REAL)) is non trivial V49() set
NORMSTR(# (product (carr G)),(zeros G),[:(addop G):],[:(multop G):],(productnorm G) #) is strict NORMSTR
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding set
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
len (carr G) is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
dom (carr G) is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
Seg (len G) is non empty V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= len G ) } is set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding set
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
F is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
F . f is set
G . f is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . f) is non empty set
dom (carr G) is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
X is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
X . f is set
(carr G) . f is set
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding set
product G is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty set
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
zeros G is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
addop G is Relation-like Function-like BinOps of carr G
[:(addop G):] is Relation-like K7((product (carr G)),(product (carr G))) -defined product (carr G) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((product (carr G)),(product (carr G))),(product (carr G))))
K7((product (carr G)),(product (carr G))) is Relation-like set
K7(K7((product (carr G)),(product (carr G))),(product (carr G))) is Relation-like set
K6(K7(K7((product (carr G)),(product (carr G))),(product (carr G)))) is set
multop G is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr G
[:(multop G):] is Relation-like K7(REAL,(product (carr G))) -defined product (carr G) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(product (carr G))),(product (carr G))))
K7(REAL,(product (carr G))) is non trivial Relation-like V49() set
K7(K7(REAL,(product (carr G))),(product (carr G))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(product (carr G))),(product (carr G)))) is non trivial V49() set
productnorm G is non empty Relation-like product (carr G) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((product (carr G)),REAL))
K7((product (carr G)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((product (carr G)),REAL)) is non trivial V49() set
NORMSTR(# (product (carr G)),(zeros G),[:(addop G):],[:(multop G):],(productnorm G) #) is strict NORMSTR
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
product G is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty set
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . F is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . F) is non empty set
f is set
X is Relation-like Function-like set
Replace (F,f,) is Relation-like Function-like set
dom (carr G) is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
m is Relation-like Function-like set
dom m is set
F .--> f is trivial Relation-like dom G -defined {F} -defined Function-like one-to-one constant V49() countable set
{F} is non empty trivial V49() V53() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded set
{F} is non empty trivial V49() V53() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(REAL)
{F} --> f is non empty Relation-like {F} -defined {f} -valued Function-like constant total quasi_total V49() countable Element of K6(K7({F},{f}))
{f} is non empty trivial V49() 1 -element countable set
K7({F},{f}) is Relation-like V49() countable set
K6(K7({F},{f})) is V49() V53() countable set
dom (F .--> f) is trivial V49() V53() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6({F})
K6({F}) is V49() V53() countable set
dom (Replace (F,f,)) is set
y0 is set
(Replace (F,f,)) . y0 is set
(carr G) . y0 is set
X . y0 is set
the non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
<* the non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR *> is non empty trivial Relation-like NAT -defined Function-like constant V49() 1 -element FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding set
[1, the non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR ] is set
{1, the non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR } is non empty V49() countable set
{{1, the non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR },{1}} is non empty V49() V53() countable set
{[1, the non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR ]} is non empty trivial Relation-like Function-like constant V49() 1 -element countable set
G is non empty trivial Relation-like NAT -defined Function-like constant V49() 1 -element FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding set
dom G is non empty trivial V49() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . F is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . F is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
f is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
product G is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty set
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
F is set
dom (carr G) is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
f is Relation-like Function-like set
dom f is set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
the epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
m is set
G . the epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . the epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G) is non empty non trivial set
i is set
Replace ( the epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G,m,) is Relation-like Function-like set
Replace ( the epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G,i,) is Relation-like Function-like set
(Replace ( the epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G,m,)) . the epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G is set
F is set
f is set
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding set
product G is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty set
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
F is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
F + f is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
the addF of (product G) is Relation-like K7( the carrier of (product G), the carrier of (product G)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)))
K7( the carrier of (product G), the carrier of (product G)) is Relation-like set
K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)) is Relation-like set
K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G))) is set
K462( the carrier of (product G), the addF of (product G),F,f) is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
m is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
i is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
X is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
len (carr G) is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
dom (carr G) is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
Seg (len G) is non empty V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= len G ) } is set
zeros G is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
addop G is Relation-like Function-like BinOps of carr G
[:(addop G):] is Relation-like K7((product (carr G)),(product (carr G))) -defined product (carr G) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((product (carr G)),(product (carr G))),(product (carr G))))
K7((product (carr G)),(product (carr G))) is Relation-like set
K7(K7((product (carr G)),(product (carr G))),(product (carr G))) is Relation-like set
K6(K7(K7((product (carr G)),(product (carr G))),(product (carr G)))) is set
multop G is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr G
[:(multop G):] is Relation-like K7(REAL,(product (carr G))) -defined product (carr G) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(product (carr G))),(product (carr G))))
K7(REAL,(product (carr G))) is non trivial Relation-like V49() set
K7(K7(REAL,(product (carr G))),(product (carr G))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(product (carr G))),(product (carr G)))) is non trivial V49() set
productnorm G is non empty Relation-like product (carr G) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((product (carr G)),REAL))
K7((product (carr G)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((product (carr G)),REAL)) is non trivial V49() set
NORMSTR(# (product (carr G)),(zeros G),[:(addop G):],[:(multop G):],(productnorm G) #) is strict NORMSTR
i0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
y0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom (carr G)
(carr G) . y0 is non empty set
K7(((carr G) . y0),((carr G) . y0)) is Relation-like set
G . y0 is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . y0) is non empty set
K7( the carrier of (G . y0), the carrier of (G . y0)) is Relation-like set
(addop G) . y0 is Relation-like K7(((carr G) . y0),((carr G) . y0)) -defined (carr G) . y0 -valued Function-like total quasi_total Element of K6(K7(K7(((carr G) . y0),((carr G) . y0)),((carr G) . y0)))
K7(K7(((carr G) . y0),((carr G) . y0)),((carr G) . y0)) is Relation-like set
K6(K7(K7(((carr G) . y0),((carr G) . y0)),((carr G) . y0))) is set
the addF of (G . y0) is Relation-like K7( the carrier of (G . y0), the carrier of (G . y0)) -defined the carrier of (G . y0) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . y0), the carrier of (G . y0)), the carrier of (G . y0)))
K7(K7( the carrier of (G . y0), the carrier of (G . y0)), the carrier of (G . y0)) is Relation-like set
K6(K7(K7( the carrier of (G . y0), the carrier of (G . y0)), the carrier of (G . y0))) is set
(G,X,i0) is Element of the carrier of (G . i0)
G . i0 is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . i0) is non empty set
(G,m,i0) is Element of the carrier of (G . i0)
(G,i,i0) is Element of the carrier of (G . i0)
(G,m,i0) + (G,i,i0) is Element of the carrier of (G . i0)
the addF of (G . i0) is Relation-like K7( the carrier of (G . i0), the carrier of (G . i0)) -defined the carrier of (G . i0) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . i0), the carrier of (G . i0)), the carrier of (G . i0)))
K7( the carrier of (G . i0), the carrier of (G . i0)) is Relation-like set
K7(K7( the carrier of (G . i0), the carrier of (G . i0)), the carrier of (G . i0)) is Relation-like set
K6(K7(K7( the carrier of (G . i0), the carrier of (G . i0)), the carrier of (G . i0))) is set
K462( the carrier of (G . i0), the addF of (G . i0),(G,m,i0),(G,i,i0)) is Element of the carrier of (G . i0)
i0 is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
y0 is set
dom i0 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below Element of K6(NAT)
s is Relation-like Function-like set
dom s is set
y1 is Relation-like Function-like set
dom y1 is set
s is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom (carr G)
(carr G) . s is non empty set
K7(((carr G) . s),((carr G) . s)) is Relation-like set
G . s is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . s) is non empty set
K7( the carrier of (G . s), the carrier of (G . s)) is Relation-like set
(addop G) . s is Relation-like K7(((carr G) . s),((carr G) . s)) -defined (carr G) . s -valued Function-like total quasi_total Element of K6(K7(K7(((carr G) . s),((carr G) . s)),((carr G) . s)))
K7(K7(((carr G) . s),((carr G) . s)),((carr G) . s)) is Relation-like set
K6(K7(K7(((carr G) . s),((carr G) . s)),((carr G) . s))) is set
the addF of (G . s) is Relation-like K7( the carrier of (G . s), the carrier of (G . s)) -defined the carrier of (G . s) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . s), the carrier of (G . s)), the carrier of (G . s)))
K7(K7( the carrier of (G . s), the carrier of (G . s)), the carrier of (G . s)) is Relation-like set
K6(K7(K7( the carrier of (G . s), the carrier of (G . s)), the carrier of (G . s))) is set
i0 . y0 is set
r is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . r is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
(G,m,r) is Element of the carrier of (G . r)
the carrier of (G . r) is non empty set
(G,i,r) is Element of the carrier of (G . r)
(G,m,r) + (G,i,r) is Element of the carrier of (G . r)
the addF of (G . r) is Relation-like K7( the carrier of (G . r), the carrier of (G . r)) -defined the carrier of (G . r) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . r), the carrier of (G . r)), the carrier of (G . r)))
K7( the carrier of (G . r), the carrier of (G . r)) is Relation-like set
K7(K7( the carrier of (G . r), the carrier of (G . r)), the carrier of (G . r)) is Relation-like set
K6(K7(K7( the carrier of (G . r), the carrier of (G . r)), the carrier of (G . r))) is set
K462( the carrier of (G . r), the addF of (G . r),(G,m,r),(G,i,r)) is Element of the carrier of (G . r)
X . y0 is set
y0 is Relation-like Function-like set
dom y0 is set
r is Relation-like Function-like set
dom r is set
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding set
product G is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty set
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
F is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f is V11() real ext-real Element of REAL
f * F is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
the Mult of (product G) is Relation-like K7(REAL, the carrier of (product G)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (product G)), the carrier of (product G)))
K7(REAL, the carrier of (product G)) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of (product G)), the carrier of (product G)) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of (product G)), the carrier of (product G))) is non trivial V49() set
K458( the Mult of (product G),f,F) is set
m is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
X is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
dom (carr G) is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
i is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
i0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom (carr G)
(carr G) . i0 is non empty set
K7(REAL,((carr G) . i0)) is non trivial Relation-like V49() set
G . i0 is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . i0) is non empty set
K7(REAL, the carrier of (G . i0)) is non trivial Relation-like V49() set
multop G is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr G
(multop G) . i0 is Relation-like K7(REAL,((carr G) . i0)) -defined (carr G) . i0 -valued Function-like total quasi_total Element of K6(K7(K7(REAL,((carr G) . i0)),((carr G) . i0)))
K7(K7(REAL,((carr G) . i0)),((carr G) . i0)) is non trivial Relation-like V49() set
K6(K7(K7(REAL,((carr G) . i0)),((carr G) . i0))) is non trivial V49() set
the Mult of (G . i0) is Relation-like K7(REAL, the carrier of (G . i0)) -defined the carrier of (G . i0) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (G . i0)), the carrier of (G . i0)))
K7(K7(REAL, the carrier of (G . i0)), the carrier of (G . i0)) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of (G . i0)), the carrier of (G . i0))) is non trivial V49() set
zeros G is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
addop G is Relation-like Function-like BinOps of carr G
[:(addop G):] is Relation-like K7((product (carr G)),(product (carr G))) -defined product (carr G) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((product (carr G)),(product (carr G))),(product (carr G))))
K7((product (carr G)),(product (carr G))) is Relation-like set
K7(K7((product (carr G)),(product (carr G))),(product (carr G))) is Relation-like set
K6(K7(K7((product (carr G)),(product (carr G))),(product (carr G)))) is set
[:(multop G):] is Relation-like K7(REAL,(product (carr G))) -defined product (carr G) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(product (carr G))),(product (carr G))))
K7(REAL,(product (carr G))) is non trivial Relation-like V49() set
K7(K7(REAL,(product (carr G))),(product (carr G))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(product (carr G))),(product (carr G)))) is non trivial V49() set
productnorm G is non empty Relation-like product (carr G) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((product (carr G)),REAL))
K7((product (carr G)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((product (carr G)),REAL)) is non trivial V49() set
NORMSTR(# (product (carr G)),(zeros G),[:(addop G):],[:(multop G):],(productnorm G) #) is strict NORMSTR
(G,X,i) is Element of the carrier of (G . i)
G . i is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . i) is non empty set
(G,m,i) is Element of the carrier of (G . i)
f * (G,m,i) is Element of the carrier of (G . i)
the Mult of (G . i) is Relation-like K7(REAL, the carrier of (G . i)) -defined the carrier of (G . i) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (G . i)), the carrier of (G . i)))
K7(REAL, the carrier of (G . i)) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of (G . i)), the carrier of (G . i)) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of (G . i)), the carrier of (G . i))) is non trivial V49() set
K458( the Mult of (G . i),f,(G,m,i)) is set
i is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
i0 is set
dom i is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below Element of K6(NAT)
r is Relation-like Function-like set
dom r is set
s is Relation-like Function-like set
dom s is set
r is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom (carr G)
(carr G) . r is non empty set
K7(REAL,((carr G) . r)) is non trivial Relation-like V49() set
G . r is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . r) is non empty set
K7(REAL, the carrier of (G . r)) is non trivial Relation-like V49() set
(multop G) . r is Relation-like K7(REAL,((carr G) . r)) -defined (carr G) . r -valued Function-like total quasi_total Element of K6(K7(K7(REAL,((carr G) . r)),((carr G) . r)))
K7(K7(REAL,((carr G) . r)),((carr G) . r)) is non trivial Relation-like V49() set
K6(K7(K7(REAL,((carr G) . r)),((carr G) . r))) is non trivial V49() set
the Mult of (G . r) is Relation-like K7(REAL, the carrier of (G . r)) -defined the carrier of (G . r) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (G . r)), the carrier of (G . r)))
K7(K7(REAL, the carrier of (G . r)), the carrier of (G . r)) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of (G . r)), the carrier of (G . r))) is non trivial V49() set
i . i0 is set
y0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . y0 is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
(G,m,y0) is Element of the carrier of (G . y0)
the carrier of (G . y0) is non empty set
f * (G,m,y0) is Element of the carrier of (G . y0)
the Mult of (G . y0) is Relation-like K7(REAL, the carrier of (G . y0)) -defined the carrier of (G . y0) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (G . y0)), the carrier of (G . y0)))
K7(REAL, the carrier of (G . y0)) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of (G . y0)), the carrier of (G . y0)) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of (G . y0)), the carrier of (G . y0))) is non trivial V49() set
K458( the Mult of (G . y0),f,(G,m,y0)) is set
X . i0 is set
i0 is Relation-like Function-like set
dom i0 is set
y0 is Relation-like Function-like set
dom y0 is set
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding set
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
product G is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
0. (product G) is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like V68( product G) countable Element of the carrier of (product G)
the carrier of (product G) is non empty set
the ZeroF of (product G) is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
F is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
dom (carr G) is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
zeros G is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
addop G is Relation-like Function-like BinOps of carr G
[:(addop G):] is Relation-like K7((product (carr G)),(product (carr G))) -defined product (carr G) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((product (carr G)),(product (carr G))),(product (carr G))))
K7((product (carr G)),(product (carr G))) is Relation-like set
K7(K7((product (carr G)),(product (carr G))),(product (carr G))) is Relation-like set
K6(K7(K7((product (carr G)),(product (carr G))),(product (carr G)))) is set
multop G is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr G
[:(multop G):] is Relation-like K7(REAL,(product (carr G))) -defined product (carr G) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(product (carr G))),(product (carr G))))
K7(REAL,(product (carr G))) is non trivial Relation-like V49() set
K7(K7(REAL,(product (carr G))),(product (carr G))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(product (carr G))),(product (carr G)))) is non trivial V49() set
productnorm G is non empty Relation-like product (carr G) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((product (carr G)),REAL))
K7((product (carr G)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((product (carr G)),REAL)) is non trivial V49() set
NORMSTR(# (product (carr G)),(zeros G),[:(addop G):],[:(multop G):],(productnorm G) #) is strict NORMSTR
f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
(G,F,f) is Element of the carrier of (G . f)
G . f is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . f) is non empty set
G . f is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
0. (G . f) is V68(G . f) Element of the carrier of (G . f)
the carrier of (G . f) is non empty set
the ZeroF of (G . f) is Element of the carrier of (G . f)
f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom (carr G)
F . f is Element of (carr G) . f
(carr G) . f is non empty set
X is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . X is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
0. (G . X) is V68(G . X) Element of the carrier of (G . X)
the carrier of (G . X) is non empty set
the ZeroF of (G . X) is Element of the carrier of (G . X)
G . f is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the ZeroF of (G . f) is Element of the carrier of (G . f)
the carrier of (G . f) is non empty set
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding set
product G is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty set
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
F is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
F - f is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
- f is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
F + (- f) is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
the addF of (product G) is Relation-like K7( the carrier of (product G), the carrier of (product G)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)))
K7( the carrier of (product G), the carrier of (product G)) is Relation-like set
K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)) is Relation-like set
K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G))) is set
K462( the carrier of (product G), the addF of (product G),F,(- f)) is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
m is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
i is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
X is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
- 1 is non empty V11() real ext-real non positive negative V132() V160() Element of INT
(- 1) * f is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
the Mult of (product G) is Relation-like K7(REAL, the carrier of (product G)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (product G)), the carrier of (product G)))
K7(REAL, the carrier of (product G)) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of (product G)), the carrier of (product G)) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of (product G)), the carrier of (product G))) is non trivial V49() set
K458( the Mult of (product G),(- 1),f) is set
F + ((- 1) * f) is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),F,((- 1) * f)) is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
y0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
(G,X,y0) is Element of the carrier of (G . y0)
G . y0 is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . y0) is non empty set
(G,m,y0) is Element of the carrier of (G . y0)
(G,i,y0) is Element of the carrier of (G . y0)
(G,m,y0) - (G,i,y0) is Element of the carrier of (G . y0)
- (G,i,y0) is Element of the carrier of (G . y0)
(G,m,y0) + (- (G,i,y0)) is Element of the carrier of (G . y0)
the addF of (G . y0) is Relation-like K7( the carrier of (G . y0), the carrier of (G . y0)) -defined the carrier of (G . y0) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . y0), the carrier of (G . y0)), the carrier of (G . y0)))
K7( the carrier of (G . y0), the carrier of (G . y0)) is Relation-like set
K7(K7( the carrier of (G . y0), the carrier of (G . y0)), the carrier of (G . y0)) is Relation-like set
K6(K7(K7( the carrier of (G . y0), the carrier of (G . y0)), the carrier of (G . y0))) is set
K462( the carrier of (G . y0), the addF of (G . y0),(G,m,y0),(- (G,i,y0))) is Element of the carrier of (G . y0)
i0 is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
(G,i0,y0) is Element of the carrier of (G . y0)
(G,m,y0) + (G,i0,y0) is Element of the carrier of (G . y0)
K462( the carrier of (G . y0), the addF of (G . y0),(G,m,y0),(G,i0,y0)) is Element of the carrier of (G . y0)
(- 1) * (G,i,y0) is Element of the carrier of (G . y0)
the Mult of (G . y0) is Relation-like K7(REAL, the carrier of (G . y0)) -defined the carrier of (G . y0) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (G . y0)), the carrier of (G . y0)))
K7(REAL, the carrier of (G . y0)) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of (G . y0)), the carrier of (G . y0)) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of (G . y0)), the carrier of (G . y0))) is non trivial V49() set
K458( the Mult of (G . y0),(- 1),(G,i,y0)) is set
i0 is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
y0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
(G,i0,y0) is Element of the carrier of (G . y0)
G . y0 is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . y0) is non empty set
(G,i,y0) is Element of the carrier of (G . y0)
(- 1) * (G,i,y0) is Element of the carrier of (G . y0)
the Mult of (G . y0) is Relation-like K7(REAL, the carrier of (G . y0)) -defined the carrier of (G . y0) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (G . y0)), the carrier of (G . y0)))
K7(REAL, the carrier of (G . y0)) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of (G . y0)), the carrier of (G . y0)) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of (G . y0)), the carrier of (G . y0))) is non trivial V49() set
K458( the Mult of (G . y0),(- 1),(G,i,y0)) is set
(G,X,y0) is Element of the carrier of (G . y0)
(G,m,y0) is Element of the carrier of (G . y0)
(G,m,y0) - (G,i,y0) is Element of the carrier of (G . y0)
- (G,i,y0) is Element of the carrier of (G . y0)
(G,m,y0) + (- (G,i,y0)) is Element of the carrier of (G . y0)
the addF of (G . y0) is Relation-like K7( the carrier of (G . y0), the carrier of (G . y0)) -defined the carrier of (G . y0) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . y0), the carrier of (G . y0)), the carrier of (G . y0)))
K7( the carrier of (G . y0), the carrier of (G . y0)) is Relation-like set
K7(K7( the carrier of (G . y0), the carrier of (G . y0)), the carrier of (G . y0)) is Relation-like set
K6(K7(K7( the carrier of (G . y0), the carrier of (G . y0)), the carrier of (G . y0))) is set
K462( the carrier of (G . y0), the addF of (G . y0),(G,m,y0),(- (G,i,y0))) is Element of the carrier of (G . y0)
(G,m,y0) + (G,i0,y0) is Element of the carrier of (G . y0)
K462( the carrier of (G . y0), the addF of (G . y0),(G,m,y0),(G,i0,y0)) is Element of the carrier of (G . y0)
G is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of G is non empty set
F is Element of the carrier of G
f is Element of the carrier of G
f - F is Element of the carrier of G
- F is Element of the carrier of G
f + (- F) is Element of the carrier of G
the addF of G is Relation-like K7( the carrier of G, the carrier of G) -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of G, the carrier of G), the carrier of G))
K7( the carrier of G, the carrier of G) is Relation-like set
K7(K7( the carrier of G, the carrier of G), the carrier of G) is Relation-like set
K6(K7(K7( the carrier of G, the carrier of G), the carrier of G)) is set
K462( the carrier of G, the addF of G,f,(- F)) is Element of the carrier of G
{ (F + (b₁ * (f - F))) where b₁ is V11() real ext-real Element of REAL : ( not b₁ <= 0 & not 1 <= b₁ ) } is set
K6( the carrier of G) is set
X is set
m is V11() real ext-real Element of REAL
m * (f - F) is Element of the carrier of G
the Mult of G is Relation-like K7(REAL, the carrier of G) -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of G), the carrier of G))
K7(REAL, the carrier of G) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of G), the carrier of G) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of G), the carrier of G)) is non trivial V49() set
K458( the Mult of G,m,(f - F)) is set
F + (m * (f - F)) is Element of the carrier of G
K462( the carrier of G, the addF of G,F,(m * (f - F))) is Element of the carrier of G
G is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of G is non empty set
G is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of G is non empty set
F is Element of the carrier of G
f is Element of the carrier of G
f - F is Element of the carrier of G
- F is Element of the carrier of G
f + (- F) is Element of the carrier of G
the addF of G is Relation-like K7( the carrier of G, the carrier of G) -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of G, the carrier of G), the carrier of G))
K7( the carrier of G, the carrier of G) is Relation-like set
K7(K7( the carrier of G, the carrier of G), the carrier of G) is Relation-like set
K6(K7(K7( the carrier of G, the carrier of G), the carrier of G)) is set
K462( the carrier of G, the addF of G,f,(- F)) is Element of the carrier of G
X is V11() real ext-real Element of REAL
X * (f - F) is Element of the carrier of G
the Mult of G is Relation-like K7(REAL, the carrier of G) -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of G), the carrier of G))
K7(REAL, the carrier of G) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of G), the carrier of G) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of G), the carrier of G)) is non trivial V49() set
K458( the Mult of G,X,(f - F)) is set
F + (X * (f - F)) is Element of the carrier of G
K462( the carrier of G, the addF of G,F,(X * (f - F))) is Element of the carrier of G
X * f is Element of the carrier of G
K458( the Mult of G,X,f) is set
X * (- F) is Element of the carrier of G
K458( the Mult of G,X,(- F)) is set
(X * f) + (X * (- F)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(X * f),(X * (- F))) is Element of the carrier of G
F + ((X * f) + (X * (- F))) is Element of the carrier of G
K462( the carrier of G, the addF of G,F,((X * f) + (X * (- F)))) is Element of the carrier of G
X * F is Element of the carrier of G
K458( the Mult of G,X,F) is set
- (X * F) is Element of the carrier of G
(X * f) + (- (X * F)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(X * f),(- (X * F))) is Element of the carrier of G
F + ((X * f) + (- (X * F))) is Element of the carrier of G
K462( the carrier of G, the addF of G,F,((X * f) + (- (X * F)))) is Element of the carrier of G
F + (- (X * F)) is Element of the carrier of G
K462( the carrier of G, the addF of G,F,(- (X * F))) is Element of the carrier of G
(F + (- (X * F))) + (X * f) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F + (- (X * F))),(X * f)) is Element of the carrier of G
1 * F is Element of the carrier of G
K458( the Mult of G,1,F) is set
(1 * F) - (X * F) is Element of the carrier of G
(1 * F) + (- (X * F)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(1 * F),(- (X * F))) is Element of the carrier of G
((1 * F) - (X * F)) + (X * f) is Element of the carrier of G
K462( the carrier of G, the addF of G,((1 * F) - (X * F)),(X * f)) is Element of the carrier of G
1 - X is V11() real ext-real Element of REAL
K38(X) is V11() real ext-real set
K36(1,K38(X)) is V11() real ext-real set
(1 - X) * F is Element of the carrier of G
K458( the Mult of G,(1 - X),F) is set
((1 - X) * F) + (X * f) is Element of the carrier of G
K462( the carrier of G, the addF of G,((1 - X) * F),(X * f)) is Element of the carrier of G
G is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of G is non empty set
F is Element of the carrier of G
f is Element of the carrier of G
(G,F,f) is Element of K6( the carrier of G)
K6( the carrier of G) is set
f - F is Element of the carrier of G
- F is Element of the carrier of G
f + (- F) is Element of the carrier of G
the addF of G is Relation-like K7( the carrier of G, the carrier of G) -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of G, the carrier of G), the carrier of G))
K7( the carrier of G, the carrier of G) is Relation-like set
K7(K7( the carrier of G, the carrier of G), the carrier of G) is Relation-like set
K6(K7(K7( the carrier of G, the carrier of G), the carrier of G)) is set
K462( the carrier of G, the addF of G,f,(- F)) is Element of the carrier of G
{ (F + (b₁ * (f - F))) where b₁ is V11() real ext-real Element of REAL : ( not b₁ <= 0 & not 1 <= b₁ ) } is set
LSeg (F,f) is Element of K6( the carrier of G)
X is set
m is V11() real ext-real Element of REAL
m * (f - F) is Element of the carrier of G
the Mult of G is Relation-like K7(REAL, the carrier of G) -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of G), the carrier of G))
K7(REAL, the carrier of G) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of G), the carrier of G) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of G), the carrier of G)) is non trivial V49() set
K458( the Mult of G,m,(f - F)) is set
F + (m * (f - F)) is Element of the carrier of G
K462( the carrier of G, the addF of G,F,(m * (f - F))) is Element of the carrier of G
1 - m is V11() real ext-real Element of REAL
K38(m) is V11() real ext-real set
K36(1,K38(m)) is V11() real ext-real set
(1 - m) * F is Element of the carrier of G
K458( the Mult of G,(1 - m),F) is set
m * f is Element of the carrier of G
K458( the Mult of G,m,f) is set
((1 - m) * F) + (m * f) is Element of the carrier of G
K462( the carrier of G, the addF of G,((1 - m) * F),(m * f)) is Element of the carrier of G
{ (((1 - b₁) * F) + (b₁ * f)) where b₁ is V11() real ext-real Element of REAL : ( 0 <= b₁ & b₁ <= 1 ) } is set
G is V11() real ext-real Element of REAL
G / 2 is V11() real ext-real Element of REAL
K39(2) is non empty V11() real ext-real positive non negative set
K37(G,K39(2)) is V11() real ext-real set
G - (G / 2) is V11() real ext-real Element of REAL
K38((G / 2)) is V11() real ext-real set
K36(G,K38((G / 2))) is V11() real ext-real set
(G / 2) - (G / 2) is V11() real ext-real Element of REAL
K36((G / 2),K38((G / 2))) is V11() real ext-real set
G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of G is non empty non trivial set
K7(REAL, the carrier of G) is non trivial Relation-like V49() set
K6(K7(REAL, the carrier of G)) is non trivial V49() set
F is Relation-like REAL -defined the carrier of G -valued Function-like Element of K6(K7(REAL, the carrier of G))
f is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
X is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
X + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
1 / (X + 1) is non empty V11() real ext-real positive non negative V160() Element of RAT
K39((X + 1)) is non empty V11() real ext-real positive non negative set
K37(1,K39((X + 1))) is non empty V11() real ext-real positive non negative set
X is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
m is V11() real ext-real set
m " is V11() real ext-real Element of REAL
i is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
i + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
(m ") + 0 is V11() real ext-real Element of REAL
1 / (m ") is V11() real ext-real Element of REAL
K39((m ")) is V11() real ext-real set
K37(1,K39((m "))) is V11() real ext-real set
1 / (i + 1) is non empty V11() real ext-real positive non negative V160() Element of RAT
K39((i + 1)) is non empty V11() real ext-real positive non negative set
K37(1,K39((i + 1))) is non empty V11() real ext-real positive non negative set
i0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
y0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
i0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
y0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
1 / (y0 + 1) is non empty V11() real ext-real positive non negative V160() Element of RAT
K39((y0 + 1)) is non empty V11() real ext-real positive non negative set
K37(1,K39((y0 + 1))) is non empty V11() real ext-real positive non negative set
1 / (i0 + 1) is non empty V11() real ext-real positive non negative V160() Element of RAT
K39((i0 + 1)) is non empty V11() real ext-real positive non negative set
K37(1,K39((i0 + 1))) is non empty V11() real ext-real positive non negative set
X . y0 is V11() real ext-real Element of REAL
(X . y0) - 0 is V11() real ext-real Element of REAL
K38(0) is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional complex-valued ext-real-valued real-valued natural-valued V49() V50() V53() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V134() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V167() Cardinal-yielding countable bounded_below bounded_above real-bounded V251() set
K36((X . y0),K38(0)) is V11() real ext-real set
abs ((X . y0) - 0) is V11() real ext-real Element of REAL
lim X is V11() real ext-real Element of REAL
m is non empty Relation-like non-empty NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K6(K7(NAT,REAL))
F /* m is non empty Relation-like NAT -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of G))
K7(NAT, the carrier of G) is non trivial Relation-like V49() set
K6(K7(NAT, the carrier of G)) is non trivial V49() set
m " is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(m ") (#) (F /* m) is non empty Relation-like NAT -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of G))
lim ((m ") (#) (F /* m)) is Element of the carrier of G
0. G is V68(G) Element of the carrier of G
the ZeroF of G is Element of the carrier of G
i is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
((m ") (#) (F /* m)) . i is Element of the carrier of G
(((m ") (#) (F /* m)) . i) - (0. G) is Element of the carrier of G
- (0. G) is Element of the carrier of G
(((m ") (#) (F /* m)) . i) + (- (0. G)) is Element of the carrier of G
the addF of G is Relation-like K7( the carrier of G, the carrier of G) -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of G, the carrier of G), the carrier of G))
K7( the carrier of G, the carrier of G) is Relation-like set
K7(K7( the carrier of G, the carrier of G), the carrier of G) is Relation-like set
K6(K7(K7( the carrier of G, the carrier of G), the carrier of G)) is set
K462( the carrier of G, the addF of G,(((m ") (#) (F /* m)) . i),(- (0. G))) is Element of the carrier of G
||.((((m ") (#) (F /* m)) . i) - (0. G)).|| is V11() real ext-real non negative Element of REAL
the normF of G is non empty Relation-like the carrier of G -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of G,REAL))
K7( the carrier of G,REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of G,REAL)) is non trivial V49() set
the normF of G . ((((m ") (#) (F /* m)) . i) - (0. G)) is V11() real ext-real Element of REAL
m . i is V11() real ext-real Element of REAL
F /. (m . i) is Element of the carrier of G
(m . i) " is V11() real ext-real Element of REAL
((m . i) ") * (F /. (m . i)) is Element of the carrier of G
the Mult of G is Relation-like K7(REAL, the carrier of G) -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of G), the carrier of G))
K7(K7(REAL, the carrier of G), the carrier of G) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of G), the carrier of G)) is non trivial V49() set
K458( the Mult of G,((m . i) "),(F /. (m . i))) is set
||.(((m . i) ") * (F /. (m . i))).|| is V11() real ext-real non negative Element of REAL
the normF of G . (((m . i) ") * (F /. (m . i))) is V11() real ext-real Element of REAL
abs ((m . i) ") is V11() real ext-real Element of REAL
||.(F /. (m . i)).|| is V11() real ext-real non negative Element of REAL
the normF of G . (F /. (m . i)) is V11() real ext-real Element of REAL
(abs ((m . i) ")) * ||.(F /. (m . i)).|| is V11() real ext-real Element of REAL
abs (m . i) is V11() real ext-real Element of REAL
||.(F /. (m . i)).|| / (abs (m . i)) is V11() real ext-real Element of REAL
K39((abs (m . i))) is V11() real ext-real set
K37(||.(F /. (m . i)).||,K39((abs (m . i)))) is V11() real ext-real set
dom F is complex-membered ext-real-membered real-membered Element of K6(REAL)
rng m is non empty complex-membered ext-real-membered real-membered Element of K6(REAL)
||.(((m ") (#) (F /* m)) . i).|| is V11() real ext-real non negative Element of REAL
the normF of G . (((m ") (#) (F /* m)) . i) is V11() real ext-real Element of REAL
(F /* m) . i is Element of the carrier of G
(m ") . i is V11() real ext-real Element of REAL
((m ") . i) * ((F /* m) . i) is Element of the carrier of G
K458( the Mult of G,((m ") . i),((F /* m) . i)) is set
||.(((m ") . i) * ((F /* m) . i)).|| is V11() real ext-real non negative Element of REAL
the normF of G . (((m ") . i) * ((F /* m) . i)) is V11() real ext-real Element of REAL
((m . i) ") * ((F /* m) . i) is Element of the carrier of G
K458( the Mult of G,((m . i) "),((F /* m) . i)) is set
||.(((m . i) ") * ((F /* m) . i)).|| is V11() real ext-real non negative Element of REAL
the normF of G . (((m . i) ") * ((F /* m) . i)) is V11() real ext-real Element of REAL
i0 is V11() real ext-real Element of REAL
f is non empty Relation-like non-empty NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K6(K7(NAT,REAL))
lim f is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional complex-valued ext-real-valued real-valued natural-valued V49() V50() V53() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V134() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V167() Cardinal-yielding countable bounded_below bounded_above real-bounded V251() Element of REAL
X is V11() real ext-real Element of REAL
m is V11() real ext-real Element of REAL
i is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
i0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
F /* f is non empty Relation-like NAT -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of G))
f " is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(f ") (#) (F /* f) is non empty Relation-like NAT -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of G))
rng f is non empty complex-membered ext-real-membered real-membered Element of K6(REAL)
y0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
((f ") (#) (F /* f)) . y0 is Element of the carrier of G
(((f ") (#) (F /* f)) . y0) - (0. G) is Element of the carrier of G
(((f ") (#) (F /* f)) . y0) + (- (0. G)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(((f ") (#) (F /* f)) . y0),(- (0. G))) is Element of the carrier of G
||.((((f ") (#) (F /* f)) . y0) - (0. G)).|| is V11() real ext-real non negative Element of REAL
the normF of G . ((((f ") (#) (F /* f)) . y0) - (0. G)) is V11() real ext-real Element of REAL
f . y0 is V11() real ext-real Element of REAL
(f . y0) - 0 is V11() real ext-real Element of REAL
K36((f . y0),K38(0)) is V11() real ext-real set
abs ((f . y0) - 0) is V11() real ext-real Element of REAL
F /. (f . y0) is Element of the carrier of G
||.(F /. (f . y0)).|| is V11() real ext-real non negative Element of REAL
the normF of G . (F /. (f . y0)) is V11() real ext-real Element of REAL
abs (f . y0) is V11() real ext-real Element of REAL
||.(F /. (f . y0)).|| / (abs (f . y0)) is V11() real ext-real Element of REAL
K39((abs (f . y0))) is V11() real ext-real set
K37(||.(F /. (f . y0)).||,K39((abs (f . y0)))) is V11() real ext-real set
(f . y0) " is V11() real ext-real Element of REAL
abs ((f . y0) ") is V11() real ext-real Element of REAL
(abs ((f . y0) ")) * ||.(F /. (f . y0)).|| is V11() real ext-real Element of REAL
((f . y0) ") * (F /. (f . y0)) is Element of the carrier of G
K458( the Mult of G,((f . y0) "),(F /. (f . y0))) is set
||.(((f . y0) ") * (F /. (f . y0))).|| is V11() real ext-real non negative Element of REAL
the normF of G . (((f . y0) ") * (F /. (f . y0))) is V11() real ext-real Element of REAL
(F /* f) . y0 is Element of the carrier of G
((f . y0) ") * ((F /* f) . y0) is Element of the carrier of G
K458( the Mult of G,((f . y0) "),((F /* f) . y0)) is set
||.(((f . y0) ") * ((F /* f) . y0)).|| is V11() real ext-real non negative Element of REAL
the normF of G . (((f . y0) ") * ((F /* f) . y0)) is V11() real ext-real Element of REAL
(f ") . y0 is V11() real ext-real Element of REAL
((f ") . y0) * ((F /* f) . y0) is Element of the carrier of G
K458( the Mult of G,((f ") . y0),((F /* f) . y0)) is set
||.(((f ") . y0) * ((F /* f) . y0)).|| is V11() real ext-real non negative Element of REAL
the normF of G . (((f ") . y0) * ((F /* f) . y0)) is V11() real ext-real Element of REAL
||.(((f ") (#) (F /* f)) . y0).|| is V11() real ext-real non negative Element of REAL
the normF of G . (((f ") (#) (F /* f)) . y0) is V11() real ext-real Element of REAL
lim ((f ") (#) (F /* f)) is Element of the carrier of G
f is V11() real ext-real Element of REAL
X is V11() real ext-real Element of REAL
G 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))
F is V11() real ext-real Element of REAL
F is V11() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
f + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
1 / (f + 1) is non empty V11() real ext-real positive non negative V160() Element of RAT
K39((f + 1)) is non empty V11() real ext-real positive non negative set
K37(1,K39((f + 1))) is non empty V11() real ext-real positive non negative set
f is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
X is V11() real ext-real set
X " is V11() real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
m + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
(X ") + 0 is V11() real ext-real Element of REAL
1 / (X ") is V11() real ext-real Element of REAL
K39((X ")) is V11() real ext-real set
K37(1,K39((X "))) is V11() real ext-real set
1 / (m + 1) is non empty V11() real ext-real positive non negative V160() Element of RAT
K39((m + 1)) is non empty V11() real ext-real positive non negative set
K37(1,K39((m + 1))) is non empty V11() real ext-real positive non negative set
i is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
i0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
i + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
i0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
1 / (i0 + 1) is non empty V11() real ext-real positive non negative V160() Element of RAT
K39((i0 + 1)) is non empty V11() real ext-real positive non negative set
K37(1,K39((i0 + 1))) is non empty V11() real ext-real positive non negative set
1 / (i + 1) is non empty V11() real ext-real positive non negative V160() Element of RAT
K39((i + 1)) is non empty V11() real ext-real positive non negative set
K37(1,K39((i + 1))) is non empty V11() real ext-real positive non negative set
f . i0 is V11() real ext-real Element of REAL
(f . i0) - 0 is V11() real ext-real Element of REAL
K38(0) is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional complex-valued ext-real-valued real-valued natural-valued V49() V50() V53() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V134() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V167() Cardinal-yielding countable bounded_below bounded_above real-bounded V251() set
K36((f . i0),K38(0)) is V11() real ext-real set
abs ((f . i0) - 0) is V11() real ext-real Element of REAL
lim f is V11() real ext-real Element of REAL
X is non empty Relation-like non-empty NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K6(K7(NAT,REAL))
X " is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
G /* X is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(X ") (#) (G /* X) is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
lim ((X ") (#) (G /* X)) is V11() real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
((X ") (#) (G /* X)) . m is V11() real ext-real Element of REAL
(((X ") (#) (G /* X)) . m) - 0 is V11() real ext-real Element of REAL
K36((((X ") (#) (G /* X)) . m),K38(0)) is V11() real ext-real set
abs ((((X ") (#) (G /* X)) . m) - 0) is V11() real ext-real Element of REAL
X . m is V11() real ext-real Element of REAL
(X . m) " is V11() real ext-real Element of REAL
G . (X . m) is V11() real ext-real Element of REAL
((X . m) ") * (G . (X . m)) is V11() real ext-real Element of REAL
abs (((X . m) ") * (G . (X . m))) is V11() real ext-real Element of REAL
abs ((X . m) ") is V11() real ext-real Element of REAL
abs (G . (X . m)) is V11() real ext-real Element of REAL
(abs ((X . m) ")) * (abs (G . (X . m))) is V11() real ext-real Element of REAL
abs (X . m) is V11() real ext-real Element of REAL
(abs (G . (X . m))) / (abs (X . m)) is V11() real ext-real Element of REAL
K39((abs (X . m))) is V11() real ext-real set
K37((abs (G . (X . m))),K39((abs (X . m)))) is V11() real ext-real set
(X ") . m is V11() real ext-real Element of REAL
(G /* X) . m is V11() real ext-real Element of REAL
((X ") . m) * ((G /* X) . m) is V11() real ext-real Element of REAL
abs (((X ") . m) * ((G /* X) . m)) is V11() real ext-real Element of REAL
((X . m) ") * ((G /* X) . m) is V11() real ext-real Element of REAL
abs (((X . m) ") * ((G /* X) . m)) is V11() real ext-real Element of REAL
i is V11() real ext-real Element of REAL
F is non empty Relation-like non-empty NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K6(K7(NAT,REAL))
lim F is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional complex-valued ext-real-valued real-valued natural-valued V49() V50() V53() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V134() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V167() Cardinal-yielding countable bounded_below bounded_above real-bounded V251() Element of REAL
f is V11() real ext-real set
X is V11() real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
i is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
i0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
F . i0 is V11() real ext-real Element of REAL
(F . i0) - 0 is V11() real ext-real Element of REAL
K36((F . i0),K38(0)) is V11() real ext-real set
abs ((F . i0) - 0) is V11() real ext-real Element of REAL
G . (F . i0) is V11() real ext-real Element of REAL
abs (G . (F . i0)) is V11() real ext-real Element of REAL
abs (F . i0) is V11() real ext-real Element of REAL
(abs (G . (F . i0))) / (abs (F . i0)) is V11() real ext-real Element of REAL
K39((abs (F . i0))) is V11() real ext-real set
K37((abs (G . (F . i0))),K39((abs (F . i0)))) is V11() real ext-real set
(F . i0) " is V11() real ext-real Element of REAL
abs ((F . i0) ") is V11() real ext-real Element of REAL
(abs ((F . i0) ")) * (abs (G . (F . i0))) is V11() real ext-real Element of REAL
((F . i0) ") * (G . (F . i0)) is V11() real ext-real Element of REAL
abs (((F . i0) ") * (G . (F . i0))) is V11() real ext-real Element of REAL
G /* F is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(G /* F) . i0 is V11() real ext-real Element of REAL
((F . i0) ") * ((G /* F) . i0) is V11() real ext-real Element of REAL
abs (((F . i0) ") * ((G /* F) . i0)) is V11() real ext-real Element of REAL
F " is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(F ") . i0 is V11() real ext-real Element of REAL
((F ") . i0) * ((G /* F) . i0) is V11() real ext-real Element of REAL
abs (((F ") . i0) * ((G /* F) . i0)) is V11() real ext-real Element of REAL
(F ") (#) (G /* F) is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
((F ") (#) (G /* F)) . i0 is V11() real ext-real Element of REAL
(((F ") (#) (G /* F)) . i0) - 0 is V11() real ext-real Element of REAL
K36((((F ") (#) (G /* F)) . i0),K38(0)) is V11() real ext-real set
abs ((((F ") (#) (G /* F)) . i0) - 0) is V11() real ext-real Element of REAL
lim ((F ") (#) (G /* F)) is V11() real ext-real Element of REAL
F is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
[.0,1.] is complex-membered ext-real-membered real-membered closed V251() Element of K6(REAL)
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( 0 <= b₁ & b₁ <= 1 ) } is set
].0,1.[ is complex-membered ext-real-membered real-membered open non left_end non right_end V251() Element of K6(REAL)
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( not b₁ <= 0 & not 1 <= b₁ ) } is set
G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of G is non empty non trivial set
K7(REAL, the carrier of G) is non trivial Relation-like V49() set
K6(K7(REAL, the carrier of G)) is non trivial V49() set
F is Relation-like REAL -defined the carrier of G -valued Function-like Element of K6(K7(REAL, the carrier of G))
dom F is complex-membered ext-real-membered real-membered Element of K6(REAL)
F | [.0,1.] is Relation-like REAL -defined [.0,1.] -defined REAL -defined the carrier of G -valued Function-like Element of K6(K7(REAL, the carrier of G))
F /. 1 is Element of the carrier of G
F /. 0 is Element of the carrier of G
(F /. 1) - (F /. 0) is Element of the carrier of G
- (F /. 0) is Element of the carrier of G
(F /. 1) + (- (F /. 0)) is Element of the carrier of G
the addF of G is Relation-like K7( the carrier of G, the carrier of G) -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of G, the carrier of G), the carrier of G))
K7( the carrier of G, the carrier of G) is Relation-like set
K7(K7( the carrier of G, the carrier of G), the carrier of G) is Relation-like set
K6(K7(K7( the carrier of G, the carrier of G), the carrier of G)) is set
K462( the carrier of G, the addF of G,(F /. 1),(- (F /. 0))) is Element of the carrier of G
||.((F /. 1) - (F /. 0)).|| is V11() real ext-real non negative Element of REAL
the normF of G is non empty Relation-like the carrier of G -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of G,REAL))
K7( the carrier of G,REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of G,REAL)) is non trivial V49() set
the normF of G . ((F /. 1) - (F /. 0)) is V11() real ext-real Element of REAL
f is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
dom f is complex-membered ext-real-membered real-membered Element of K6(REAL)
f | [.0,1.] is Relation-like REAL -defined [.0,1.] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
f /. 1 is V11() real ext-real Element of REAL
f /. 0 is V11() real ext-real Element of REAL
(f /. 1) - (f /. 0) is V11() real ext-real Element of REAL
K38((f /. 0)) is V11() real ext-real set
K36((f /. 1),K38((f /. 0))) is V11() real ext-real set
X is V11() real ext-real Element of REAL
X / 2 is V11() real ext-real Element of REAL
K39(2) is non empty V11() real ext-real positive non negative set
K37(X,K39(2)) is V11() real ext-real set
f . 0 is V11() real ext-real Element of REAL
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( b₁ in [.0,1.] & ((||.((F /. b₁) - (F /. 0)).|| - ((f . b₁) - (f . 0))) - ((X / 2) * b₁)) - (X / 2) <= 0 ) } is set
i0 is set
y0 is V11() real ext-real Element of REAL
F /. y0 is Element of the carrier of G
(F /. y0) - (F /. 0) is Element of the carrier of G
(F /. y0) + (- (F /. 0)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. y0),(- (F /. 0))) is Element of the carrier of G
||.((F /. y0) - (F /. 0)).|| is V11() real ext-real non negative Element of REAL
the normF of G . ((F /. y0) - (F /. 0)) is V11() real ext-real Element of REAL
f . y0 is V11() real ext-real Element of REAL
(f . y0) - (f . 0) is V11() real ext-real Element of REAL
K38((f . 0)) is V11() real ext-real set
K36((f . y0),K38((f . 0))) is V11() real ext-real set
||.((F /. y0) - (F /. 0)).|| - ((f . y0) - (f . 0)) is V11() real ext-real Element of REAL
K38(((f . y0) - (f . 0))) is V11() real ext-real set
K36(||.((F /. y0) - (F /. 0)).||,K38(((f . y0) - (f . 0)))) is V11() real ext-real set
(X / 2) * y0 is V11() real ext-real Element of REAL
(||.((F /. y0) - (F /. 0)).|| - ((f . y0) - (f . 0))) - ((X / 2) * y0) is V11() real ext-real Element of REAL
K38(((X / 2) * y0)) is V11() real ext-real set
K36((||.((F /. y0) - (F /. 0)).|| - ((f . y0) - (f . 0))),K38(((X / 2) * y0))) is V11() real ext-real set
((||.((F /. y0) - (F /. 0)).|| - ((f . y0) - (f . 0))) - ((X / 2) * y0)) - (X / 2) is V11() real ext-real Element of REAL
K38((X / 2)) is V11() real ext-real set
K36(((||.((F /. y0) - (F /. 0)).|| - ((f . y0) - (f . 0))) - ((X / 2) * y0)),K38((X / 2))) is V11() real ext-real set
y0 is V11() real ext-real set
i0 is complex-membered ext-real-membered real-membered Element of K6(REAL)
r is V11() real ext-real Element of REAL
F /. r is Element of the carrier of G
(F /. r) - (F /. 0) is Element of the carrier of G
(F /. r) + (- (F /. 0)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. r),(- (F /. 0))) is Element of the carrier of G
||.((F /. r) - (F /. 0)).|| is V11() real ext-real non negative Element of REAL
the normF of G . ((F /. r) - (F /. 0)) is V11() real ext-real Element of REAL
f . r is V11() real ext-real Element of REAL
(f . r) - (f . 0) is V11() real ext-real Element of REAL
K38((f . 0)) is V11() real ext-real set
K36((f . r),K38((f . 0))) is V11() real ext-real set
||.((F /. r) - (F /. 0)).|| - ((f . r) - (f . 0)) is V11() real ext-real Element of REAL
K38(((f . r) - (f . 0))) is V11() real ext-real set
K36(||.((F /. r) - (F /. 0)).||,K38(((f . r) - (f . 0)))) is V11() real ext-real set
(X / 2) * r is V11() real ext-real Element of REAL
(||.((F /. r) - (F /. 0)).|| - ((f . r) - (f . 0))) - ((X / 2) * r) is V11() real ext-real Element of REAL
K38(((X / 2) * r)) is V11() real ext-real set
K36((||.((F /. r) - (F /. 0)).|| - ((f . r) - (f . 0))),K38(((X / 2) * r))) is V11() real ext-real set
((||.((F /. r) - (F /. 0)).|| - ((f . r) - (f . 0))) - ((X / 2) * r)) - (X / 2) is V11() real ext-real Element of REAL
K38((X / 2)) is V11() real ext-real set
K36(((||.((F /. r) - (F /. 0)).|| - ((f . r) - (f . 0))) - ((X / 2) * r)),K38((X / 2))) is V11() real ext-real set
abs y0 is V11() real ext-real Element of REAL
r is V11() real ext-real Element of REAL
r is V11() real ext-real Element of REAL
upper_bound i0 is V11() real ext-real Element of REAL
r is V11() real ext-real set
r / 2 is V11() real ext-real Element of REAL
K37(r,K39(2)) is V11() real ext-real set
min ((r / 2),1) is V11() real ext-real Element of REAL
y1 is V11() real ext-real Element of REAL
y1 - 0 is V11() real ext-real Element of REAL
K38(0) is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional complex-valued ext-real-valued real-valued natural-valued V49() V50() V53() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V134() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V167() Cardinal-yielding countable bounded_below bounded_above real-bounded V251() set
K36(y1,K38(0)) is V11() real ext-real set
abs (y1 - 0) is V11() real ext-real Element of REAL
F /. y1 is Element of the carrier of G
(F /. y1) - (F /. 0) is Element of the carrier of G
(F /. y1) + (- (F /. 0)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. y1),(- (F /. 0))) is Element of the carrier of G
||.((F /. y1) - (F /. 0)).|| is V11() real ext-real non negative Element of REAL
the normF of G . ((F /. y1) - (F /. 0)) is V11() real ext-real Element of REAL
[.0,y1.] is complex-membered ext-real-membered real-membered closed V251() Element of K6(REAL)
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( 0 <= b₁ & b₁ <= y1 ) } is set
f | [.0,y1.] is Relation-like REAL -defined [.0,y1.] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
].0,y1.[ is complex-membered ext-real-membered real-membered open non left_end non right_end V251() Element of K6(REAL)
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( not b₁ <= 0 & not y1 <= b₁ ) } is set
f . y1 is V11() real ext-real Element of REAL
(f . y1) - (f . 0) is V11() real ext-real Element of REAL
K38((f . 0)) is V11() real ext-real set
K36((f . y1),K38((f . 0))) is V11() real ext-real set
((f . y1) - (f . 0)) / (y1 - 0) is V11() real ext-real Element of REAL
K39((y1 - 0)) is V11() real ext-real set
K37(((f . y1) - (f . 0)),K39((y1 - 0))) is V11() real ext-real set
PP is V11() real ext-real Element of REAL
diff (f,PP) is V11() real ext-real Element of REAL
diff (F,PP) is Element of the carrier of G
||.(diff (F,PP)).|| is V11() real ext-real non negative Element of REAL
the normF of G . (diff (F,PP)) is V11() real ext-real Element of REAL
0 + ||.((F /. y1) - (F /. 0)).|| is V11() real ext-real non negative Element of REAL
((f . y1) - (f . 0)) + (X / 2) is V11() real ext-real Element of REAL
(X / 2) * y1 is V11() real ext-real Element of REAL
(((f . y1) - (f . 0)) + (X / 2)) + ((X / 2) * y1) is V11() real ext-real Element of REAL
||.((F /. y1) - (F /. 0)).|| - ((((f . y1) - (f . 0)) + (X / 2)) + ((X / 2) * y1)) is V11() real ext-real Element of REAL
K38(((((f . y1) - (f . 0)) + (X / 2)) + ((X / 2) * y1))) is V11() real ext-real set
K36(||.((F /. y1) - (F /. 0)).||,K38(((((f . y1) - (f . 0)) + (X / 2)) + ((X / 2) * y1)))) is V11() real ext-real set
||.((F /. y1) - (F /. 0)).|| - ((f . y1) - (f . 0)) is V11() real ext-real Element of REAL
K38(((f . y1) - (f . 0))) is V11() real ext-real set
K36(||.((F /. y1) - (F /. 0)).||,K38(((f . y1) - (f . 0)))) is V11() real ext-real set
(||.((F /. y1) - (F /. 0)).|| - ((f . y1) - (f . 0))) - ((X / 2) * y1) is V11() real ext-real Element of REAL
K38(((X / 2) * y1)) is V11() real ext-real set
K36((||.((F /. y1) - (F /. 0)).|| - ((f . y1) - (f . 0))),K38(((X / 2) * y1))) is V11() real ext-real set
((||.((F /. y1) - (F /. 0)).|| - ((f . y1) - (f . 0))) - ((X / 2) * y1)) - (X / 2) is V11() real ext-real Element of REAL
K38((X / 2)) is V11() real ext-real set
K36(((||.((F /. y1) - (F /. 0)).|| - ((f . y1) - (f . 0))) - ((X / 2) * y1)),K38((X / 2))) is V11() real ext-real set
r is V11() real ext-real set
r is V11() real ext-real set
s is V11() real ext-real Element of REAL
F /. s is Element of the carrier of G
(F /. s) - (F /. 0) is Element of the carrier of G
(F /. s) + (- (F /. 0)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. s),(- (F /. 0))) is Element of the carrier of G
||.((F /. s) - (F /. 0)).|| is V11() real ext-real non negative Element of REAL
the normF of G . ((F /. s) - (F /. 0)) is V11() real ext-real Element of REAL
f . s is V11() real ext-real Element of REAL
(f . s) - (f . 0) is V11() real ext-real Element of REAL
K38((f . 0)) is V11() real ext-real set
K36((f . s),K38((f . 0))) is V11() real ext-real set
||.((F /. s) - (F /. 0)).|| - ((f . s) - (f . 0)) is V11() real ext-real Element of REAL
K38(((f . s) - (f . 0))) is V11() real ext-real set
K36(||.((F /. s) - (F /. 0)).||,K38(((f . s) - (f . 0)))) is V11() real ext-real set
(X / 2) * s is V11() real ext-real Element of REAL
(||.((F /. s) - (F /. 0)).|| - ((f . s) - (f . 0))) - ((X / 2) * s) is V11() real ext-real Element of REAL
K38(((X / 2) * s)) is V11() real ext-real set
K36((||.((F /. s) - (F /. 0)).|| - ((f . s) - (f . 0))),K38(((X / 2) * s))) is V11() real ext-real set
((||.((F /. s) - (F /. 0)).|| - ((f . s) - (f . 0))) - ((X / 2) * s)) - (X / 2) is V11() real ext-real Element of REAL
K38((X / 2)) is V11() real ext-real set
K36(((||.((F /. s) - (F /. 0)).|| - ((f . s) - (f . 0))) - ((X / 2) * s)),K38((X / 2))) is V11() real ext-real set
s is V11() real ext-real Element of REAL
r is V11() real ext-real set
r is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
1 + r is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
1 / (1 + r) is non empty V11() real ext-real positive non negative V160() Element of RAT
K39((1 + r)) is non empty V11() real ext-real positive non negative set
K37(1,K39((1 + r))) is non empty V11() real ext-real positive non negative set
s is V11() real ext-real set
(upper_bound i0) - s is V11() real ext-real Element of REAL
K38(s) is V11() real ext-real set
K36((upper_bound i0),K38(s)) is V11() real ext-real set
y1 is V11() real ext-real set
y1 is V11() real ext-real set
PP is V11() real ext-real Element of REAL
DD is V11() real ext-real Element of REAL
DD + s is V11() real ext-real Element of REAL
((upper_bound i0) - s) + s is V11() real ext-real Element of REAL
s + DD is V11() real ext-real Element of REAL
(s + DD) - DD is V11() real ext-real Element of REAL
K38(DD) is V11() real ext-real set
K36((s + DD),K38(DD)) is V11() real ext-real set
(upper_bound i0) - DD is V11() real ext-real Element of REAL
K36((upper_bound i0),K38(DD)) is V11() real ext-real set
abs ((upper_bound i0) - DD) is V11() real ext-real Element of REAL
r + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
1 / (r + 1) is non empty V11() real ext-real positive non negative V160() Element of RAT
K39((r + 1)) is non empty V11() real ext-real positive non negative set
K37(1,K39((r + 1))) is non empty V11() real ext-real positive non negative set
r is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
y1 is V11() real ext-real set
y1 / 2 is V11() real ext-real Element of REAL
K37(y1,K39(2)) is V11() real ext-real set
1 / (y1 / 2) is V11() real ext-real Element of REAL
K39((y1 / 2)) is V11() real ext-real set
K37(1,K39((y1 / 2))) is V11() real ext-real set
DD is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
DD + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
(1 / (y1 / 2)) + 0 is V11() real ext-real Element of REAL
1 / (DD + 1) is non empty V11() real ext-real positive non negative V160() Element of RAT
K39((DD + 1)) is non empty V11() real ext-real positive non negative set
K37(1,K39((DD + 1))) is non empty V11() real ext-real positive non negative set
1 / (1 / (y1 / 2)) is V11() real ext-real Element of REAL
K39((1 / (y1 / 2))) is V11() real ext-real set
K37(1,K39((1 / (y1 / 2)))) is V11() real ext-real set
a is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
s is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
t is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
s . t is V11() real ext-real Element of REAL
(s . t) - (upper_bound i0) is V11() real ext-real Element of REAL
K38((upper_bound i0)) is V11() real ext-real set
K36((s . t),K38((upper_bound i0))) is V11() real ext-real set
abs ((s . t) - (upper_bound i0)) is V11() real ext-real Element of REAL
a + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
t + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
1 / (t + 1) is non empty V11() real ext-real positive non negative V160() Element of RAT
K39((t + 1)) is non empty V11() real ext-real positive non negative set
K37(1,K39((t + 1))) is non empty V11() real ext-real positive non negative set
1 / (a + 1) is non empty V11() real ext-real positive non negative V160() Element of RAT
K39((a + 1)) is non empty V11() real ext-real positive non negative set
K37(1,K39((a + 1))) is non empty V11() real ext-real positive non negative set
(upper_bound i0) - (s . t) is V11() real ext-real Element of REAL
K38((s . t)) is V11() real ext-real set
K36((upper_bound i0),K38((s . t))) is V11() real ext-real set
abs ((upper_bound i0) - (s . t)) is V11() real ext-real Element of REAL
1 + t is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
1 / (1 + t) is non empty V11() real ext-real positive non negative V160() Element of RAT
K39((1 + t)) is non empty V11() real ext-real positive non negative set
K37(1,K39((1 + t))) is non empty V11() real ext-real positive non negative set
lim s is V11() real ext-real Element of REAL
y1 is V11() real ext-real Element of REAL
F /. y1 is Element of the carrier of G
(F /. y1) - (F /. 0) is Element of the carrier of G
(F /. y1) + (- (F /. 0)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. y1),(- (F /. 0))) is Element of the carrier of G
||.((F /. y1) - (F /. 0)).|| is V11() real ext-real non negative Element of REAL
the normF of G . ((F /. y1) - (F /. 0)) is V11() real ext-real Element of REAL
f . y1 is V11() real ext-real Element of REAL
(f . y1) - (f . 0) is V11() real ext-real Element of REAL
K38((f . 0)) is V11() real ext-real set
K36((f . y1),K38((f . 0))) is V11() real ext-real set
||.((F /. y1) - (F /. 0)).|| - ((f . y1) - (f . 0)) is V11() real ext-real Element of REAL
K38(((f . y1) - (f . 0))) is V11() real ext-real set
K36(||.((F /. y1) - (F /. 0)).||,K38(((f . y1) - (f . 0)))) is V11() real ext-real set
(X / 2) * y1 is V11() real ext-real Element of REAL
(||.((F /. y1) - (F /. 0)).|| - ((f . y1) - (f . 0))) - ((X / 2) * y1) is V11() real ext-real Element of REAL
K38(((X / 2) * y1)) is V11() real ext-real set
K36((||.((F /. y1) - (F /. 0)).|| - ((f . y1) - (f . 0))),K38(((X / 2) * y1))) is V11() real ext-real set
((||.((F /. y1) - (F /. 0)).|| - ((f . y1) - (f . 0))) - ((X / 2) * y1)) - (X / 2) is V11() real ext-real Element of REAL
K38((X / 2)) is V11() real ext-real set
K36(((||.((F /. y1) - (F /. 0)).|| - ((f . y1) - (f . 0))) - ((X / 2) * y1)),K38((X / 2))) is V11() real ext-real set
y1 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))
y1 | [.0,1.] is Relation-like REAL -defined [.0,1.] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
dom y1 is non empty complex-membered ext-real-membered real-membered Element of K6(REAL)
dom (y1 | [.0,1.]) is complex-membered ext-real-membered real-membered Element of K6(REAL)
DD is set
rng s is non empty complex-membered ext-real-membered real-membered Element of K6(REAL)
a is set
s . a is V11() real ext-real Element of REAL
a is V11() real ext-real Element of REAL
F /. a is Element of the carrier of G
(F /. a) - (F /. 0) is Element of the carrier of G
(F /. a) + (- (F /. 0)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. a),(- (F /. 0))) is Element of the carrier of G
||.((F /. a) - (F /. 0)).|| is V11() real ext-real non negative Element of REAL
the normF of G . ((F /. a) - (F /. 0)) is V11() real ext-real Element of REAL
f . a is V11() real ext-real Element of REAL
(f . a) - (f . 0) is V11() real ext-real Element of REAL
K38((f . 0)) is V11() real ext-real set
K36((f . a),K38((f . 0))) is V11() real ext-real set
||.((F /. a) - (F /. 0)).|| - ((f . a) - (f . 0)) is V11() real ext-real Element of REAL
K38(((f . a) - (f . 0))) is V11() real ext-real set
K36(||.((F /. a) - (F /. 0)).||,K38(((f . a) - (f . 0)))) is V11() real ext-real set
(X / 2) * a is V11() real ext-real Element of REAL
(||.((F /. a) - (F /. 0)).|| - ((f . a) - (f . 0))) - ((X / 2) * a) is V11() real ext-real Element of REAL
K38(((X / 2) * a)) is V11() real ext-real set
K36((||.((F /. a) - (F /. 0)).|| - ((f . a) - (f . 0))),K38(((X / 2) * a))) is V11() real ext-real set
((||.((F /. a) - (F /. 0)).|| - ((f . a) - (f . 0))) - ((X / 2) * a)) - (X / 2) is V11() real ext-real Element of REAL
K38((X / 2)) is V11() real ext-real set
K36(((||.((F /. a) - (F /. 0)).|| - ((f . a) - (f . 0))) - ((X / 2) * a)),K38((X / 2))) is V11() real ext-real set
DD is V11() real ext-real set
DD / 3 is V11() real ext-real Element of REAL
K39(3) is non empty V11() real ext-real positive non negative set
K37(DD,K39(3)) is V11() real ext-real set
F /. (upper_bound i0) is Element of the carrier of G
t is V11() real ext-real set
f . (upper_bound i0) is V11() real ext-real Element of REAL
tm is V11() real ext-real set
(DD / 3) / (X / 2) is V11() real ext-real Element of REAL
K39((X / 2)) is V11() real ext-real set
K37((DD / 3),K39((X / 2))) is V11() real ext-real set
((DD / 3) / (X / 2)) / 2 is V11() real ext-real Element of REAL
K37(((DD / 3) / (X / 2)),K39(2)) is V11() real ext-real set
((DD / 3) / (X / 2)) * (X / 2) is V11() real ext-real Element of REAL
(X / 2) * (((DD / 3) / (X / 2)) / 2) is V11() real ext-real Element of REAL
min (t,tm) is V11() real ext-real set
min ((min (t,tm)),(((DD / 3) / (X / 2)) / 2)) is V11() real ext-real set
ys is V11() real ext-real set
(y1 | [.0,1.]) . (upper_bound i0) is V11() real ext-real Element of REAL
rm is V11() real ext-real set
rm - (upper_bound i0) is V11() real ext-real Element of REAL
K38((upper_bound i0)) is V11() real ext-real set
K36(rm,K38((upper_bound i0))) is V11() real ext-real set
abs (rm - (upper_bound i0)) is V11() real ext-real Element of REAL
(y1 | [.0,1.]) . rm is V11() real ext-real Element of REAL
((y1 | [.0,1.]) . rm) - ((y1 | [.0,1.]) . (upper_bound i0)) is V11() real ext-real Element of REAL
K38(((y1 | [.0,1.]) . (upper_bound i0))) is V11() real ext-real set
K36(((y1 | [.0,1.]) . rm),K38(((y1 | [.0,1.]) . (upper_bound i0)))) is V11() real ext-real set
abs (((y1 | [.0,1.]) . rm) - ((y1 | [.0,1.]) . (upper_bound i0))) is V11() real ext-real Element of REAL
y1 . rm is V11() real ext-real Element of REAL
F /. rm is Element of the carrier of G
(F /. rm) - (F /. 0) is Element of the carrier of G
(F /. rm) + (- (F /. 0)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. rm),(- (F /. 0))) is Element of the carrier of G
||.((F /. rm) - (F /. 0)).|| is V11() real ext-real non negative Element of REAL
the normF of G . ((F /. rm) - (F /. 0)) is V11() real ext-real Element of REAL
f . rm is V11() real ext-real Element of REAL
(f . rm) - (f . 0) is V11() real ext-real Element of REAL
K38((f . 0)) is V11() real ext-real set
K36((f . rm),K38((f . 0))) is V11() real ext-real set
||.((F /. rm) - (F /. 0)).|| - ((f . rm) - (f . 0)) is V11() real ext-real Element of REAL
K38(((f . rm) - (f . 0))) is V11() real ext-real set
K36(||.((F /. rm) - (F /. 0)).||,K38(((f . rm) - (f . 0)))) is V11() real ext-real set
(X / 2) * rm is V11() real ext-real Element of REAL
(||.((F /. rm) - (F /. 0)).|| - ((f . rm) - (f . 0))) - ((X / 2) * rm) is V11() real ext-real Element of REAL
K38(((X / 2) * rm)) is V11() real ext-real set
K36((||.((F /. rm) - (F /. 0)).|| - ((f . rm) - (f . 0))),K38(((X / 2) * rm))) is V11() real ext-real set
((||.((F /. rm) - (F /. 0)).|| - ((f . rm) - (f . 0))) - ((X / 2) * rm)) - (X / 2) is V11() real ext-real Element of REAL
K38((X / 2)) is V11() real ext-real set
K36(((||.((F /. rm) - (F /. 0)).|| - ((f . rm) - (f . 0))) - ((X / 2) * rm)),K38((X / 2))) is V11() real ext-real set
y1 . (upper_bound i0) is V11() real ext-real Element of REAL
(F /. (upper_bound i0)) - (F /. 0) is Element of the carrier of G
(F /. (upper_bound i0)) + (- (F /. 0)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. (upper_bound i0)),(- (F /. 0))) is Element of the carrier of G
||.((F /. (upper_bound i0)) - (F /. 0)).|| is V11() real ext-real non negative Element of REAL
the normF of G . ((F /. (upper_bound i0)) - (F /. 0)) is V11() real ext-real Element of REAL
(f . (upper_bound i0)) - (f . 0) is V11() real ext-real Element of REAL
K36((f . (upper_bound i0)),K38((f . 0))) is V11() real ext-real set
||.((F /. (upper_bound i0)) - (F /. 0)).|| - ((f . (upper_bound i0)) - (f . 0)) is V11() real ext-real Element of REAL
K38(((f . (upper_bound i0)) - (f . 0))) is V11() real ext-real set
K36(||.((F /. (upper_bound i0)) - (F /. 0)).||,K38(((f . (upper_bound i0)) - (f . 0)))) is V11() real ext-real set
(X / 2) * (upper_bound i0) is V11() real ext-real Element of REAL
(||.((F /. (upper_bound i0)) - (F /. 0)).|| - ((f . (upper_bound i0)) - (f . 0))) - ((X / 2) * (upper_bound i0)) is V11() real ext-real Element of REAL
K38(((X / 2) * (upper_bound i0))) is V11() real ext-real set
K36((||.((F /. (upper_bound i0)) - (F /. 0)).|| - ((f . (upper_bound i0)) - (f . 0))),K38(((X / 2) * (upper_bound i0)))) is V11() real ext-real set
((||.((F /. (upper_bound i0)) - (F /. 0)).|| - ((f . (upper_bound i0)) - (f . 0))) - ((X / 2) * (upper_bound i0))) - (X / 2) is V11() real ext-real Element of REAL
K36(((||.((F /. (upper_bound i0)) - (F /. 0)).|| - ((f . (upper_bound i0)) - (f . 0))) - ((X / 2) * (upper_bound i0))),K38((X / 2))) is V11() real ext-real set
||.((F /. rm) - (F /. 0)).|| - ||.((F /. (upper_bound i0)) - (F /. 0)).|| is V11() real ext-real Element of REAL
K38(||.((F /. (upper_bound i0)) - (F /. 0)).||) is V11() real ext-real non positive set
K36(||.((F /. rm) - (F /. 0)).||,K38(||.((F /. (upper_bound i0)) - (F /. 0)).||)) is V11() real ext-real set
(f . rm) - (f . (upper_bound i0)) is V11() real ext-real Element of REAL
K38((f . (upper_bound i0))) is V11() real ext-real set
K36((f . rm),K38((f . (upper_bound i0)))) is V11() real ext-real set
(||.((F /. rm) - (F /. 0)).|| - ||.((F /. (upper_bound i0)) - (F /. 0)).||) - ((f . rm) - (f . (upper_bound i0))) is V11() real ext-real Element of REAL
K38(((f . rm) - (f . (upper_bound i0)))) is V11() real ext-real set
K36((||.((F /. rm) - (F /. 0)).|| - ||.((F /. (upper_bound i0)) - (F /. 0)).||),K38(((f . rm) - (f . (upper_bound i0))))) is V11() real ext-real set
(X / 2) * (rm - (upper_bound i0)) is V11() real ext-real Element of REAL
((||.((F /. rm) - (F /. 0)).|| - ||.((F /. (upper_bound i0)) - (F /. 0)).||) - ((f . rm) - (f . (upper_bound i0)))) - ((X / 2) * (rm - (upper_bound i0))) is V11() real ext-real Element of REAL
K38(((X / 2) * (rm - (upper_bound i0)))) is V11() real ext-real set
K36(((||.((F /. rm) - (F /. 0)).|| - ||.((F /. (upper_bound i0)) - (F /. 0)).||) - ((f . rm) - (f . (upper_bound i0)))),K38(((X / 2) * (rm - (upper_bound i0))))) is V11() real ext-real set
abs ((||.((F /. rm) - (F /. 0)).|| - ||.((F /. (upper_bound i0)) - (F /. 0)).||) - ((f . rm) - (f . (upper_bound i0)))) is V11() real ext-real Element of REAL
abs ((X / 2) * (rm - (upper_bound i0))) is V11() real ext-real Element of REAL
(abs ((||.((F /. rm) - (F /. 0)).|| - ||.((F /. (upper_bound i0)) - (F /. 0)).||) - ((f . rm) - (f . (upper_bound i0))))) + (abs ((X / 2) * (rm - (upper_bound i0)))) is V11() real ext-real Element of REAL
abs (||.((F /. rm) - (F /. 0)).|| - ||.((F /. (upper_bound i0)) - (F /. 0)).||) is V11() real ext-real Element of REAL
abs ((f . rm) - (f . (upper_bound i0))) is V11() real ext-real Element of REAL
(abs (||.((F /. rm) - (F /. 0)).|| - ||.((F /. (upper_bound i0)) - (F /. 0)).||)) + (abs ((f . rm) - (f . (upper_bound i0)))) is V11() real ext-real Element of REAL
((abs (||.((F /. rm) - (F /. 0)).|| - ||.((F /. (upper_bound i0)) - (F /. 0)).||)) + (abs ((f . rm) - (f . (upper_bound i0))))) + (abs ((X / 2) * (rm - (upper_bound i0)))) is V11() real ext-real Element of REAL
((F /. rm) - (F /. 0)) - ((F /. (upper_bound i0)) - (F /. 0)) is Element of the carrier of G
- ((F /. (upper_bound i0)) - (F /. 0)) is Element of the carrier of G
((F /. rm) - (F /. 0)) + (- ((F /. (upper_bound i0)) - (F /. 0))) is Element of the carrier of G
K462( the carrier of G, the addF of G,((F /. rm) - (F /. 0)),(- ((F /. (upper_bound i0)) - (F /. 0)))) is Element of the carrier of G
||.(((F /. rm) - (F /. 0)) - ((F /. (upper_bound i0)) - (F /. 0))).|| is V11() real ext-real non negative Element of REAL
the normF of G . (((F /. rm) - (F /. 0)) - ((F /. (upper_bound i0)) - (F /. 0))) is V11() real ext-real Element of REAL
(F /. 0) - (- ((F /. (upper_bound i0)) - (F /. 0))) is Element of the carrier of G
- (- ((F /. (upper_bound i0)) - (F /. 0))) is Element of the carrier of G
(F /. 0) + (- (- ((F /. (upper_bound i0)) - (F /. 0)))) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. 0),(- (- ((F /. (upper_bound i0)) - (F /. 0))))) is Element of the carrier of G
(F /. rm) - ((F /. 0) - (- ((F /. (upper_bound i0)) - (F /. 0)))) is Element of the carrier of G
- ((F /. 0) - (- ((F /. (upper_bound i0)) - (F /. 0)))) is Element of the carrier of G
(F /. rm) + (- ((F /. 0) - (- ((F /. (upper_bound i0)) - (F /. 0))))) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. rm),(- ((F /. 0) - (- ((F /. (upper_bound i0)) - (F /. 0)))))) is Element of the carrier of G
(F /. 0) + ((F /. (upper_bound i0)) - (F /. 0)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. 0),((F /. (upper_bound i0)) - (F /. 0))) is Element of the carrier of G
(F /. rm) - ((F /. 0) + ((F /. (upper_bound i0)) - (F /. 0))) is Element of the carrier of G
- ((F /. 0) + ((F /. (upper_bound i0)) - (F /. 0))) is Element of the carrier of G
(F /. rm) + (- ((F /. 0) + ((F /. (upper_bound i0)) - (F /. 0)))) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. rm),(- ((F /. 0) + ((F /. (upper_bound i0)) - (F /. 0))))) is Element of the carrier of G
(F /. 0) - (F /. 0) is Element of the carrier of G
(F /. 0) + (- (F /. 0)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. 0),(- (F /. 0))) is Element of the carrier of G
(F /. (upper_bound i0)) - ((F /. 0) - (F /. 0)) is Element of the carrier of G
- ((F /. 0) - (F /. 0)) is Element of the carrier of G
(F /. (upper_bound i0)) + (- ((F /. 0) - (F /. 0))) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. (upper_bound i0)),(- ((F /. 0) - (F /. 0)))) is Element of the carrier of G
(F /. rm) - ((F /. (upper_bound i0)) - ((F /. 0) - (F /. 0))) is Element of the carrier of G
- ((F /. (upper_bound i0)) - ((F /. 0) - (F /. 0))) is Element of the carrier of G
(F /. rm) + (- ((F /. (upper_bound i0)) - ((F /. 0) - (F /. 0)))) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. rm),(- ((F /. (upper_bound i0)) - ((F /. 0) - (F /. 0))))) is Element of the carrier of G
0. G is V68(G) Element of the carrier of G
the ZeroF of G is Element of the carrier of G
(F /. (upper_bound i0)) - (0. G) is Element of the carrier of G
- (0. G) is Element of the carrier of G
(F /. (upper_bound i0)) + (- (0. G)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. (upper_bound i0)),(- (0. G))) is Element of the carrier of G
(F /. rm) - ((F /. (upper_bound i0)) - (0. G)) is Element of the carrier of G
- ((F /. (upper_bound i0)) - (0. G)) is Element of the carrier of G
(F /. rm) + (- ((F /. (upper_bound i0)) - (0. G))) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. rm),(- ((F /. (upper_bound i0)) - (0. G)))) is Element of the carrier of G
(F /. rm) - (F /. (upper_bound i0)) is Element of the carrier of G
- (F /. (upper_bound i0)) is Element of the carrier of G
(F /. rm) + (- (F /. (upper_bound i0))) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. rm),(- (F /. (upper_bound i0)))) is Element of the carrier of G
||.((F /. rm) - (F /. (upper_bound i0))).|| is V11() real ext-real non negative Element of REAL
the normF of G . ((F /. rm) - (F /. (upper_bound i0))) is V11() real ext-real Element of REAL
||.((F /. rm) - (F /. (upper_bound i0))).|| + (abs ((f . rm) - (f . (upper_bound i0)))) is V11() real ext-real Element of REAL
(||.((F /. rm) - (F /. (upper_bound i0))).|| + (abs ((f . rm) - (f . (upper_bound i0))))) + (abs ((X / 2) * (rm - (upper_bound i0)))) is V11() real ext-real Element of REAL
||.((F /. rm) - (F /. (upper_bound i0))).|| + (DD / 3) is V11() real ext-real Element of REAL
(abs (rm - (upper_bound i0))) * (X / 2) is V11() real ext-real Element of REAL
(((DD / 3) / (X / 2)) / 2) * (X / 2) is V11() real ext-real Element of REAL
abs (X / 2) is V11() real ext-real Element of REAL
(abs (rm - (upper_bound i0))) * (abs (X / 2)) is V11() real ext-real Element of REAL
(DD / 3) + (DD / 3) is V11() real ext-real Element of REAL
((DD / 3) + (DD / 3)) + (DD / 3) is V11() real ext-real Element of REAL
(y1 | [.0,1.]) /* s is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
lim ((y1 | [.0,1.]) /* s) is V11() real ext-real Element of REAL
- ((y1 | [.0,1.]) /* s) is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
DD is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
(- ((y1 | [.0,1.]) /* s)) . DD is V11() real ext-real Element of REAL
((y1 | [.0,1.]) /* s) . DD is V11() real ext-real Element of REAL
- (((y1 | [.0,1.]) /* s) . DD) is V11() real ext-real Element of REAL
s . DD is V11() real ext-real Element of REAL
(y1 | [.0,1.]) . (s . DD) is V11() real ext-real Element of REAL
- ((y1 | [.0,1.]) . (s . DD)) is V11() real ext-real Element of REAL
(upper_bound i0) - (s . DD) is V11() real ext-real Element of REAL
K38((s . DD)) is V11() real ext-real set
K36((upper_bound i0),K38((s . DD))) is V11() real ext-real set
abs ((upper_bound i0) - (s . DD)) is V11() real ext-real Element of REAL
DD + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
1 / (DD + 1) is non empty V11() real ext-real positive non negative V160() Element of RAT
K39((DD + 1)) is non empty V11() real ext-real positive non negative set
K37(1,K39((DD + 1))) is non empty V11() real ext-real positive non negative set
y1 . (s . DD) is V11() real ext-real Element of REAL
a is V11() real ext-real Element of REAL
F /. a is Element of the carrier of G
(F /. a) - (F /. 0) is Element of the carrier of G
(F /. a) + (- (F /. 0)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. a),(- (F /. 0))) is Element of the carrier of G
||.((F /. a) - (F /. 0)).|| is V11() real ext-real non negative Element of REAL
the normF of G . ((F /. a) - (F /. 0)) is V11() real ext-real Element of REAL
f . a is V11() real ext-real Element of REAL
(f . a) - (f . 0) is V11() real ext-real Element of REAL
K38((f . 0)) is V11() real ext-real set
K36((f . a),K38((f . 0))) is V11() real ext-real set
||.((F /. a) - (F /. 0)).|| - ((f . a) - (f . 0)) is V11() real ext-real Element of REAL
K38(((f . a) - (f . 0))) is V11() real ext-real set
K36(||.((F /. a) - (F /. 0)).||,K38(((f . a) - (f . 0)))) is V11() real ext-real set
(X / 2) * a is V11() real ext-real Element of REAL
(||.((F /. a) - (F /. 0)).|| - ((f . a) - (f . 0))) - ((X / 2) * a) is V11() real ext-real Element of REAL
K38(((X / 2) * a)) is V11() real ext-real set
K36((||.((F /. a) - (F /. 0)).|| - ((f . a) - (f . 0))),K38(((X / 2) * a))) is V11() real ext-real set
((||.((F /. a) - (F /. 0)).|| - ((f . a) - (f . 0))) - ((X / 2) * a)) - (X / 2) is V11() real ext-real Element of REAL
K38((X / 2)) is V11() real ext-real set
K36(((||.((F /. a) - (F /. 0)).|| - ((f . a) - (f . 0))) - ((X / 2) * a)),K38((X / 2))) is V11() real ext-real set
a is V11() real ext-real Element of REAL
F /. a is Element of the carrier of G
(F /. a) - (F /. 0) is Element of the carrier of G
(F /. a) + (- (F /. 0)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. a),(- (F /. 0))) is Element of the carrier of G
||.((F /. a) - (F /. 0)).|| is V11() real ext-real non negative Element of REAL
the normF of G . ((F /. a) - (F /. 0)) is V11() real ext-real Element of REAL
f . a is V11() real ext-real Element of REAL
(f . a) - (f . 0) is V11() real ext-real Element of REAL
K38((f . 0)) is V11() real ext-real set
K36((f . a),K38((f . 0))) is V11() real ext-real set
||.((F /. a) - (F /. 0)).|| - ((f . a) - (f . 0)) is V11() real ext-real Element of REAL
K38(((f . a) - (f . 0))) is V11() real ext-real set
K36(||.((F /. a) - (F /. 0)).||,K38(((f . a) - (f . 0)))) is V11() real ext-real set
(X / 2) * a is V11() real ext-real Element of REAL
(||.((F /. a) - (F /. 0)).|| - ((f . a) - (f . 0))) - ((X / 2) * a) is V11() real ext-real Element of REAL
K38(((X / 2) * a)) is V11() real ext-real set
K36((||.((F /. a) - (F /. 0)).|| - ((f . a) - (f . 0))),K38(((X / 2) * a))) is V11() real ext-real set
((||.((F /. a) - (F /. 0)).|| - ((f . a) - (f . 0))) - ((X / 2) * a)) - (X / 2) is V11() real ext-real Element of REAL
K38((X / 2)) is V11() real ext-real set
K36(((||.((F /. a) - (F /. 0)).|| - ((f . a) - (f . 0))) - ((X / 2) * a)),K38((X / 2))) is V11() real ext-real set
lim (- ((y1 | [.0,1.]) /* s)) is V11() real ext-real Element of REAL
- (lim ((y1 | [.0,1.]) /* s)) is V11() real ext-real Element of REAL
y1 . (upper_bound i0) is V11() real ext-real Element of REAL
(F /. (upper_bound i0)) - (F /. 0) is Element of the carrier of G
(F /. (upper_bound i0)) + (- (F /. 0)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. (upper_bound i0)),(- (F /. 0))) is Element of the carrier of G
||.((F /. (upper_bound i0)) - (F /. 0)).|| is V11() real ext-real non negative Element of REAL
the normF of G . ((F /. (upper_bound i0)) - (F /. 0)) is V11() real ext-real Element of REAL
(f . (upper_bound i0)) - (f . 0) is V11() real ext-real Element of REAL
K38((f . 0)) is V11() real ext-real set
K36((f . (upper_bound i0)),K38((f . 0))) is V11() real ext-real set
||.((F /. (upper_bound i0)) - (F /. 0)).|| - ((f . (upper_bound i0)) - (f . 0)) is V11() real ext-real Element of REAL
K38(((f . (upper_bound i0)) - (f . 0))) is V11() real ext-real set
K36(||.((F /. (upper_bound i0)) - (F /. 0)).||,K38(((f . (upper_bound i0)) - (f . 0)))) is V11() real ext-real set
(X / 2) * (upper_bound i0) is V11() real ext-real Element of REAL
(||.((F /. (upper_bound i0)) - (F /. 0)).|| - ((f . (upper_bound i0)) - (f . 0))) - ((X / 2) * (upper_bound i0)) is V11() real ext-real Element of REAL
K38(((X / 2) * (upper_bound i0))) is V11() real ext-real set
K36((||.((F /. (upper_bound i0)) - (F /. 0)).|| - ((f . (upper_bound i0)) - (f . 0))),K38(((X / 2) * (upper_bound i0)))) is V11() real ext-real set
((||.((F /. (upper_bound i0)) - (F /. 0)).|| - ((f . (upper_bound i0)) - (f . 0))) - ((X / 2) * (upper_bound i0))) - (X / 2) is V11() real ext-real Element of REAL
K38((X / 2)) is V11() real ext-real set
K36(((||.((F /. (upper_bound i0)) - (F /. 0)).|| - ((f . (upper_bound i0)) - (f . 0))) - ((X / 2) * (upper_bound i0))),K38((X / 2))) is V11() real ext-real set
diff (F,(upper_bound i0)) is Element of the carrier of G
DD is complex-membered ext-real-membered real-membered open Neighbourhood of upper_bound i0
a is non empty Relation-like REAL -defined the carrier of G -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of G))
a . 1 is Element of the carrier of G
t is Relation-like REAL -defined the carrier of G -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of G))
a is non empty Relation-like REAL -defined the carrier of G -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of G))
a . 1 is Element of the carrier of G
t is Relation-like REAL -defined the carrier of G -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of G))
diff (f,(upper_bound i0)) is V11() real ext-real Element of REAL
tm is complex-membered ext-real-membered real-membered open Neighbourhood of upper_bound i0
Sw1 is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued linear Element of K6(K7(REAL,REAL))
Sw1 . 1 is V11() real ext-real Element of REAL
w2 is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued RestFunc-like Element of K6(K7(REAL,REAL))
Sw1 is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued linear Element of K6(K7(REAL,REAL))
Sw1 . 1 is V11() real ext-real Element of REAL
w2 is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued RestFunc-like Element of K6(K7(REAL,REAL))
ys is complex-membered ext-real-membered real-membered open Neighbourhood of upper_bound i0
rm is V11() real ext-real set
(upper_bound i0) - rm is V11() real ext-real Element of REAL
K38(rm) is V11() real ext-real set
K36((upper_bound i0),K38(rm)) is V11() real ext-real set
(upper_bound i0) + rm is V11() real ext-real Element of REAL
].((upper_bound i0) - rm),((upper_bound i0) + rm).[ is complex-membered ext-real-membered real-membered open non left_end non right_end V251() Element of K6(REAL)
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( not b₁ <= (upper_bound i0) - rm & not (upper_bound i0) + rm <= b₁ ) } is set
rs is complex-membered ext-real-membered real-membered open Neighbourhood of upper_bound i0
i is complex-membered ext-real-membered real-membered open Neighbourhood of upper_bound i0
p1 is V11() real ext-real set
(upper_bound i0) - p1 is V11() real ext-real Element of REAL
K38(p1) is V11() real ext-real set
K36((upper_bound i0),K38(p1)) is V11() real ext-real set
(upper_bound i0) + p1 is V11() real ext-real Element of REAL
].((upper_bound i0) - p1),((upper_bound i0) + p1).[ is complex-membered ext-real-membered real-membered open non left_end non right_end V251() Element of K6(REAL)
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( not b₁ <= (upper_bound i0) - p1 & not (upper_bound i0) + p1 <= b₁ ) } is set
(X / 2) / 2 is V11() real ext-real Element of REAL
K37((X / 2),K39(2)) is V11() real ext-real set
P1 is V11() real ext-real Element of REAL
P0 is V11() real ext-real Element of REAL
min (p1,P1) is V11() real ext-real set
min ((min (p1,P1)),P0) is V11() real ext-real set
(min ((min (p1,P1)),P0)) / 2 is V11() real ext-real Element of REAL
K37((min ((min (p1,P1)),P0)),K39(2)) is V11() real ext-real set
(upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2) is V11() real ext-real Element of REAL
F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2)) is Element of the carrier of G
(F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (F /. (upper_bound i0)) is Element of the carrier of G
- (F /. (upper_bound i0)) is Element of the carrier of G
(F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) + (- (F /. (upper_bound i0))) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))),(- (F /. (upper_bound i0)))) is Element of the carrier of G
((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2)) - (upper_bound i0) is V11() real ext-real Element of REAL
K38((upper_bound i0)) is V11() real ext-real set
K36(((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2)),K38((upper_bound i0))) is V11() real ext-real set
a . (((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2)) - (upper_bound i0)) is Element of the carrier of G
t /. (((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2)) - (upper_bound i0)) is Element of the carrier of G
(a . (((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2)) - (upper_bound i0))) + (t /. (((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2)) - (upper_bound i0))) is Element of the carrier of G
K462( the carrier of G, the addF of G,(a . (((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2)) - (upper_bound i0))),(t /. (((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2)) - (upper_bound i0)))) is Element of the carrier of G
f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2)) is V11() real ext-real Element of REAL
(f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (f . (upper_bound i0)) is V11() real ext-real Element of REAL
K38((f . (upper_bound i0))) is V11() real ext-real set
K36((f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))),K38((f . (upper_bound i0)))) is V11() real ext-real set
Sw1 . (((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2)) - (upper_bound i0)) is V11() real ext-real Element of REAL
w2 . (((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2)) - (upper_bound i0)) is V11() real ext-real Element of REAL
(Sw1 . (((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2)) - (upper_bound i0))) + (w2 . (((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2)) - (upper_bound i0))) is V11() real ext-real Element of REAL
g is Element of the carrier of G
1 * g is Element of the carrier of G
the Mult of G is Relation-like K7(REAL, the carrier of G) -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of G), the carrier of G))
K7(K7(REAL, the carrier of G), the carrier of G) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of G), the carrier of G)) is non trivial V49() set
K458( the Mult of G,1,g) is set
a . ((min ((min (p1,P1)),P0)) / 2) is Element of the carrier of G
((min ((min (p1,P1)),P0)) / 2) * (diff (F,(upper_bound i0))) is Element of the carrier of G
K458( the Mult of G,((min ((min (p1,P1)),P0)) / 2),(diff (F,(upper_bound i0)))) is set
Z is V11() real ext-real Element of REAL
Z * 1 is V11() real ext-real Element of REAL
Sw1 . ((min ((min (p1,P1)),P0)) / 2) is V11() real ext-real Element of REAL
((min ((min (p1,P1)),P0)) / 2) * (diff (f,(upper_bound i0))) is V11() real ext-real Element of REAL
||.((F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (F /. (upper_bound i0))).|| is V11() real ext-real non negative Element of REAL
the normF of G . ((F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (F /. (upper_bound i0))) is V11() real ext-real Element of REAL
||.(a . ((min ((min (p1,P1)),P0)) / 2)).|| is V11() real ext-real non negative Element of REAL
the normF of G . (a . ((min ((min (p1,P1)),P0)) / 2)) is V11() real ext-real Element of REAL
t /. ((min ((min (p1,P1)),P0)) / 2) is Element of the carrier of G
||.(t /. ((min ((min (p1,P1)),P0)) / 2)).|| is V11() real ext-real non negative Element of REAL
the normF of G . (t /. ((min ((min (p1,P1)),P0)) / 2)) is V11() real ext-real Element of REAL
||.(a . ((min ((min (p1,P1)),P0)) / 2)).|| + ||.(t /. ((min ((min (p1,P1)),P0)) / 2)).|| is V11() real ext-real non negative Element of REAL
abs ((min ((min (p1,P1)),P0)) / 2) is V11() real ext-real Element of REAL
||.(diff (F,(upper_bound i0))).|| is V11() real ext-real non negative Element of REAL
the normF of G . (diff (F,(upper_bound i0))) is V11() real ext-real Element of REAL
(abs ((min ((min (p1,P1)),P0)) / 2)) * ||.(diff (F,(upper_bound i0))).|| is V11() real ext-real Element of REAL
||.(diff (F,(upper_bound i0))).|| * ((min ((min (p1,P1)),P0)) / 2) is V11() real ext-real Element of REAL
||.(t /. ((min ((min (p1,P1)),P0)) / 2)).|| / (abs ((min ((min (p1,P1)),P0)) / 2)) is V11() real ext-real Element of REAL
K39((abs ((min ((min (p1,P1)),P0)) / 2))) is V11() real ext-real set
K37(||.(t /. ((min ((min (p1,P1)),P0)) / 2)).||,K39((abs ((min ((min (p1,P1)),P0)) / 2)))) is V11() real ext-real set
((X / 2) / 2) * (abs ((min ((min (p1,P1)),P0)) / 2)) is V11() real ext-real Element of REAL
((X / 2) / 2) * ((min ((min (p1,P1)),P0)) / 2) is V11() real ext-real Element of REAL
(||.(diff (F,(upper_bound i0))).|| * ((min ((min (p1,P1)),P0)) / 2)) + (((X / 2) / 2) * ((min ((min (p1,P1)),P0)) / 2)) is V11() real ext-real Element of REAL
(diff (f,(upper_bound i0))) * ((min ((min (p1,P1)),P0)) / 2) is V11() real ext-real Element of REAL
((diff (f,(upper_bound i0))) * ((min ((min (p1,P1)),P0)) / 2)) + (((X / 2) / 2) * ((min ((min (p1,P1)),P0)) / 2)) is V11() real ext-real Element of REAL
w2 . ((min ((min (p1,P1)),P0)) / 2) is V11() real ext-real Element of REAL
abs (w2 . ((min ((min (p1,P1)),P0)) / 2)) is V11() real ext-real Element of REAL
(abs (w2 . ((min ((min (p1,P1)),P0)) / 2))) / (abs ((min ((min (p1,P1)),P0)) / 2)) is V11() real ext-real Element of REAL
K37((abs (w2 . ((min ((min (p1,P1)),P0)) / 2))),K39((abs ((min ((min (p1,P1)),P0)) / 2)))) is V11() real ext-real set
- (((X / 2) / 2) * ((min ((min (p1,P1)),P0)) / 2)) is V11() real ext-real Element of REAL
(((min ((min (p1,P1)),P0)) / 2) * (diff (f,(upper_bound i0)))) - (((X / 2) / 2) * ((min ((min (p1,P1)),P0)) / 2)) is V11() real ext-real Element of REAL
K38((((X / 2) / 2) * ((min ((min (p1,P1)),P0)) / 2))) is V11() real ext-real set
K36((((min ((min (p1,P1)),P0)) / 2) * (diff (f,(upper_bound i0)))),K38((((X / 2) / 2) * ((min ((min (p1,P1)),P0)) / 2)))) is V11() real ext-real set
((f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (f . (upper_bound i0))) + (((X / 2) / 2) * ((min ((min (p1,P1)),P0)) / 2)) is V11() real ext-real Element of REAL
(((f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (f . (upper_bound i0))) + (((X / 2) / 2) * ((min ((min (p1,P1)),P0)) / 2))) + (((X / 2) / 2) * ((min ((min (p1,P1)),P0)) / 2)) is V11() real ext-real Element of REAL
(X / 2) * ((min ((min (p1,P1)),P0)) / 2) is V11() real ext-real Element of REAL
((f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (f . (upper_bound i0))) + ((X / 2) * ((min ((min (p1,P1)),P0)) / 2)) is V11() real ext-real Element of REAL
||.((F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (F /. (upper_bound i0))).|| - (((f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (f . (upper_bound i0))) + ((X / 2) * ((min ((min (p1,P1)),P0)) / 2))) is V11() real ext-real Element of REAL
K38((((f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (f . (upper_bound i0))) + ((X / 2) * ((min ((min (p1,P1)),P0)) / 2)))) is V11() real ext-real set
K36(||.((F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (F /. (upper_bound i0))).||,K38((((f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (f . (upper_bound i0))) + ((X / 2) * ((min ((min (p1,P1)),P0)) / 2))))) is V11() real ext-real set
||.((F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (F /. (upper_bound i0))).|| - (f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) is V11() real ext-real Element of REAL
K38((f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2)))) is V11() real ext-real set
K36(||.((F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (F /. (upper_bound i0))).||,K38((f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))))) is V11() real ext-real set
(||.((F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (F /. (upper_bound i0))).|| - (f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2)))) + (f . (upper_bound i0)) is V11() real ext-real Element of REAL
((||.((F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (F /. (upper_bound i0))).|| - (f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2)))) + (f . (upper_bound i0))) - ((X / 2) * ((min ((min (p1,P1)),P0)) / 2)) is V11() real ext-real Element of REAL
K38(((X / 2) * ((min ((min (p1,P1)),P0)) / 2))) is V11() real ext-real set
K36(((||.((F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (F /. (upper_bound i0))).|| - (f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2)))) + (f . (upper_bound i0))),K38(((X / 2) * ((min ((min (p1,P1)),P0)) / 2)))) is V11() real ext-real set
(((||.((F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (F /. (upper_bound i0))).|| - (f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2)))) + (f . (upper_bound i0))) - ((X / 2) * ((min ((min (p1,P1)),P0)) / 2))) + (((||.((F /. (upper_bound i0)) - (F /. 0)).|| - ((f . (upper_bound i0)) - (f . 0))) - ((X / 2) * (upper_bound i0))) - (X / 2)) is V11() real ext-real Element of REAL
0 + 0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional complex-valued ext-real-valued real-valued natural-valued V49() V50() V53() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V134() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V167() Cardinal-yielding countable bounded_below bounded_above real-bounded V251() Element of REAL
(F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (F /. 0) is Element of the carrier of G
(F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) + (- (F /. 0)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))),(- (F /. 0))) is Element of the carrier of G
||.((F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (F /. 0)).|| is V11() real ext-real non negative Element of REAL
the normF of G . ((F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (F /. 0)) is V11() real ext-real Element of REAL
||.((F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (F /. (upper_bound i0))).|| + ||.((F /. (upper_bound i0)) - (F /. 0)).|| is V11() real ext-real non negative Element of REAL
(f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (f . 0) is V11() real ext-real Element of REAL
K36((f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))),K38((f . 0))) is V11() real ext-real set
((min ((min (p1,P1)),P0)) / 2) + (upper_bound i0) is V11() real ext-real Element of REAL
(X / 2) * (((min ((min (p1,P1)),P0)) / 2) + (upper_bound i0)) is V11() real ext-real Element of REAL
((f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (f . 0)) + ((X / 2) * (((min ((min (p1,P1)),P0)) / 2) + (upper_bound i0))) is V11() real ext-real Element of REAL
(((f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (f . 0)) + ((X / 2) * (((min ((min (p1,P1)),P0)) / 2) + (upper_bound i0)))) + (X / 2) is V11() real ext-real Element of REAL
||.((F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (F /. 0)).|| - ((((f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (f . 0)) + ((X / 2) * (((min ((min (p1,P1)),P0)) / 2) + (upper_bound i0)))) + (X / 2)) is V11() real ext-real Element of REAL
K38(((((f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (f . 0)) + ((X / 2) * (((min ((min (p1,P1)),P0)) / 2) + (upper_bound i0)))) + (X / 2))) is V11() real ext-real set
K36(||.((F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (F /. 0)).||,K38(((((f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (f . 0)) + ((X / 2) * (((min ((min (p1,P1)),P0)) / 2) + (upper_bound i0)))) + (X / 2)))) is V11() real ext-real set
(||.((F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (F /. (upper_bound i0))).|| + ||.((F /. (upper_bound i0)) - (F /. 0)).||) - ((((f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (f . 0)) + ((X / 2) * (((min ((min (p1,P1)),P0)) / 2) + (upper_bound i0)))) + (X / 2)) is V11() real ext-real Element of REAL
K36((||.((F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (F /. (upper_bound i0))).|| + ||.((F /. (upper_bound i0)) - (F /. 0)).||),K38(((((f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (f . 0)) + ((X / 2) * (((min ((min (p1,P1)),P0)) / 2) + (upper_bound i0)))) + (X / 2)))) is V11() real ext-real set
||.((F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (F /. 0)).|| - ((f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (f . 0)) is V11() real ext-real Element of REAL
K38(((f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (f . 0))) is V11() real ext-real set
K36(||.((F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (F /. 0)).||,K38(((f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (f . 0)))) is V11() real ext-real set
(X / 2) * ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2)) is V11() real ext-real Element of REAL
(||.((F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (F /. 0)).|| - ((f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (f . 0))) - ((X / 2) * ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) is V11() real ext-real Element of REAL
K38(((X / 2) * ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2)))) is V11() real ext-real set
K36((||.((F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (F /. 0)).|| - ((f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (f . 0))),K38(((X / 2) * ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))))) is V11() real ext-real set
((||.((F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (F /. 0)).|| - ((f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (f . 0))) - ((X / 2) * ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2)))) - (X / 2) is V11() real ext-real Element of REAL
K36(((||.((F /. ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (F /. 0)).|| - ((f . ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) - (f . 0))) - ((X / 2) * ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2)))),K38((X / 2))) is V11() real ext-real set
1 - 0 is non empty V11() real ext-real positive non negative Element of REAL
K38(0) is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional complex-valued ext-real-valued real-valued natural-valued V49() V50() V53() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V134() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V167() Cardinal-yielding countable bounded_below bounded_above real-bounded V251() set
K36(1,K38(0)) is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal countable set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal countable Element of REAL
2 * ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2)) is V11() real ext-real Element of REAL
(0 + 1) - (2 * ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))) is V11() real ext-real Element of REAL
K38((2 * ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2)))) is V11() real ext-real set
K36((0 + 1),K38((2 * ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2))))) is V11() real ext-real set
abs ((0 + 1) - (2 * ((upper_bound i0) + ((min ((min (p1,P1)),P0)) / 2)))) is V11() real ext-real Element of REAL
(upper_bound i0) + 0 is V11() real ext-real Element of REAL
f . 1 is V11() real ext-real Element of REAL
||.((F /. 1) - (F /. 0)).|| - ((f /. 1) - (f /. 0)) is V11() real ext-real Element of REAL
K38(((f /. 1) - (f /. 0))) is V11() real ext-real set
K36(||.((F /. 1) - (F /. 0)).||,K38(((f /. 1) - (f /. 0)))) is V11() real ext-real set
(||.((F /. 1) - (F /. 0)).|| - ((f /. 1) - (f /. 0))) - X is V11() real ext-real Element of REAL
K38(X) is V11() real ext-real set
K36((||.((F /. 1) - (F /. 0)).|| - ((f /. 1) - (f /. 0))),K38(X)) is V11() real ext-real set
((||.((F /. 1) - (F /. 0)).|| - ((f /. 1) - (f /. 0))) - X) + X is V11() real ext-real Element of REAL
0 + X is V11() real ext-real Element of REAL
(||.((F /. 1) - (F /. 0)).|| - ((f /. 1) - (f /. 0))) + ((f /. 1) - (f /. 0)) is V11() real ext-real Element of REAL
0 + ((f /. 1) - (f /. 0)) is V11() real ext-real Element of REAL
G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of G is non empty non trivial set
F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of F is non empty non trivial set
K7( the carrier of G, the carrier of F) is Relation-like set
K6(K7( the carrier of G, the carrier of F)) is set
R_NormSpace_of_BoundedLinearOperators (G,F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators (G,F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators (G,F)))
R_VectorSpace_of_LinearOperators (G,F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators (G,F) is non empty functional Element of K6( the carrier of K266( the carrier of G,F))
K266( the carrier of G,F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of G, the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of G, the carrier of F
FuncZero ( the carrier of G,F) is Relation-like the carrier of G -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of G, the carrier of F)
0. F is V68(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
the carrier of G --> (0. F) is non empty Relation-like the carrier of G -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of G, the carrier of F))
FuncAdd ( the carrier of G,F) is Relation-like K7((Funcs ( the carrier of G, the carrier of F)),(Funcs ( the carrier of G, the carrier of F))) -defined Funcs ( the carrier of G, the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of G, the carrier of F)),(Funcs ( the carrier of G, the carrier of F))),(Funcs ( the carrier of G, the carrier of F))))
K7((Funcs ( the carrier of G, the carrier of F)),(Funcs ( the carrier of G, the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of G, the carrier of F)),(Funcs ( the carrier of G, the carrier of F))),(Funcs ( the carrier of G, the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of G, the carrier of F)),(Funcs ( the carrier of G, the carrier of F))),(Funcs ( the carrier of G, the carrier of F)))) is set
FuncExtMult ( the carrier of G,F) is Relation-like K7(REAL,(Funcs ( the carrier of G, the carrier of F))) -defined Funcs ( the carrier of G, the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of G, the carrier of F))),(Funcs ( the carrier of G, the carrier of F))))
K7(REAL,(Funcs ( the carrier of G, the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of G, the carrier of F))),(Funcs ( the carrier of G, the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of G, the carrier of F))),(Funcs ( the carrier of G, the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of G, the carrier of F)),(FuncZero ( the carrier of G,F)),(FuncAdd ( the carrier of G,F)),(FuncExtMult ( the carrier of G,F)) #) is strict RLSStruct
the carrier of K266( the carrier of G,F) is non empty set
K6( the carrier of K266( the carrier of G,F)) is set
K215(K266( the carrier of G,F),(LinearOperators (G,F))) is Relation-like Function-like Element of LinearOperators (G,F)
K213(K266( the carrier of G,F),(LinearOperators (G,F))) is Relation-like K7((LinearOperators (G,F)),(LinearOperators (G,F))) -defined LinearOperators (G,F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators (G,F)),(LinearOperators (G,F))),(LinearOperators (G,F))))
K7((LinearOperators (G,F)),(LinearOperators (G,F))) is Relation-like set
K7(K7((LinearOperators (G,F)),(LinearOperators (G,F))),(LinearOperators (G,F))) is Relation-like set
K6(K7(K7((LinearOperators (G,F)),(LinearOperators (G,F))),(LinearOperators (G,F)))) is set
K214(K266( the carrier of G,F),(LinearOperators (G,F))) is Relation-like K7(REAL,(LinearOperators (G,F))) -defined LinearOperators (G,F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators (G,F))),(LinearOperators (G,F))))
K7(REAL,(LinearOperators (G,F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators (G,F))),(LinearOperators (G,F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators (G,F))),(LinearOperators (G,F)))) is non trivial V49() set
RLSStruct(# (LinearOperators (G,F)),K215(K266( the carrier of G,F),(LinearOperators (G,F))),K213(K266( the carrier of G,F),(LinearOperators (G,F))),K214(K266( the carrier of G,F),(LinearOperators (G,F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators (G,F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators (G,F))) is set
K215((R_VectorSpace_of_LinearOperators (G,F)),(BoundedLinearOperators (G,F))) is Element of BoundedLinearOperators (G,F)
K213((R_VectorSpace_of_LinearOperators (G,F)),(BoundedLinearOperators (G,F))) is Relation-like K7((BoundedLinearOperators (G,F)),(BoundedLinearOperators (G,F))) -defined BoundedLinearOperators (G,F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators (G,F)),(BoundedLinearOperators (G,F))),(BoundedLinearOperators (G,F))))
K7((BoundedLinearOperators (G,F)),(BoundedLinearOperators (G,F))) is Relation-like set
K7(K7((BoundedLinearOperators (G,F)),(BoundedLinearOperators (G,F))),(BoundedLinearOperators (G,F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators (G,F)),(BoundedLinearOperators (G,F))),(BoundedLinearOperators (G,F)))) is set
K214((R_VectorSpace_of_LinearOperators (G,F)),(BoundedLinearOperators (G,F))) is Relation-like K7(REAL,(BoundedLinearOperators (G,F))) -defined BoundedLinearOperators (G,F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators (G,F))),(BoundedLinearOperators (G,F))))
K7(REAL,(BoundedLinearOperators (G,F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators (G,F))),(BoundedLinearOperators (G,F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators (G,F))),(BoundedLinearOperators (G,F)))) is non trivial V49() set
BoundedLinearOperatorsNorm (G,F) is non empty Relation-like BoundedLinearOperators (G,F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators (G,F)),REAL))
K7((BoundedLinearOperators (G,F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators (G,F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators (G,F)),K215((R_VectorSpace_of_LinearOperators (G,F)),(BoundedLinearOperators (G,F))),K213((R_VectorSpace_of_LinearOperators (G,F)),(BoundedLinearOperators (G,F))),K214((R_VectorSpace_of_LinearOperators (G,F)),(BoundedLinearOperators (G,F))),(BoundedLinearOperatorsNorm (G,F)) #) is strict NORMSTR
f is Relation-like the carrier of G -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of G, the carrier of F))
dom f is Element of K6( the carrier of G)
K6( the carrier of G) is set
X is Element of the carrier of G
m is Element of the carrier of G
LSeg (X,m) is Element of K6( the carrier of G)
(G,X,m) is Element of K6( the carrier of G)
m - X is Element of the carrier of G
- X is Element of the carrier of G
m + (- X) is Element of the carrier of G
the addF of G is Relation-like K7( the carrier of G, the carrier of G) -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of G, the carrier of G), the carrier of G))
K7( the carrier of G, the carrier of G) is Relation-like set
K7(K7( the carrier of G, the carrier of G), the carrier of G) is Relation-like set
K6(K7(K7( the carrier of G, the carrier of G), the carrier of G)) is set
K462( the carrier of G, the addF of G,m,(- X)) is Element of the carrier of G
{ (X + (b₁ * (m - X))) where b₁ is V11() real ext-real Element of REAL : ( not b₁ <= 0 & not 1 <= b₁ ) } is set
f /. m is Element of the carrier of F
f /. X is Element of the carrier of F
(f /. m) - (f /. X) is Element of the carrier of F
- (f /. X) is Element of the carrier of F
(f /. m) + (- (f /. X)) is Element of the carrier of F
the addF of F is Relation-like K7( the carrier of F, the carrier of F) -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of F, the carrier of F), the carrier of F))
K7( the carrier of F, the carrier of F) is Relation-like set
K7(K7( the carrier of F, the carrier of F), the carrier of F) is Relation-like set
K6(K7(K7( the carrier of F, the carrier of F), the carrier of F)) is set
K462( the carrier of F, the addF of F,(f /. m),(- (f /. X))) is Element of the carrier of F
||.((f /. m) - (f /. X)).|| is V11() real ext-real non negative Element of REAL
the normF of F is non empty Relation-like the carrier of F -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of F,REAL))
K7( the carrier of F,REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of F,REAL)) is non trivial V49() set
the normF of F . ((f /. m) - (f /. X)) is V11() real ext-real Element of REAL
||.(m - X).|| is V11() real ext-real non negative Element of REAL
the normF of G is non empty Relation-like the carrier of G -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of G,REAL))
K7( the carrier of G,REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of G,REAL)) is non trivial V49() set
the normF of G . (m - X) is V11() real ext-real Element of REAL
i is V11() real ext-real Element of REAL
i * ||.(m - X).|| is V11() real ext-real Element of REAL
K7(REAL, the carrier of G) is non trivial Relation-like V49() set
K6(K7(REAL, the carrier of G)) is non trivial V49() set
i0 is non empty Relation-like REAL -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(REAL, the carrier of G))
i0 | [.0,1.] is Relation-like REAL -defined [.0,1.] -defined REAL -defined the carrier of G -valued Function-like Element of K6(K7(REAL, the carrier of G))
dom i0 is non empty complex-membered ext-real-membered real-membered Element of K6(REAL)
dom (i0 | [.0,1.]) is complex-membered ext-real-membered real-membered Element of K6(REAL)
r is V11() real ext-real set
i0 /. r is Element of the carrier of G
i0 . r is set
r * (m - X) is Element of the carrier of G
the Mult of G is Relation-like K7(REAL, the carrier of G) -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of G), the carrier of G))
K7(K7(REAL, the carrier of G), the carrier of G) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of G), the carrier of G)) is non trivial V49() set
K458( the Mult of G,r,(m - X)) is set
(r * (m - X)) + X is Element of the carrier of G
K462( the carrier of G, the addF of G,(r * (m - X)),X) is Element of the carrier of G
r is V11() real ext-real Element of REAL
(i0 | [.0,1.]) /. r is Element of the carrier of G
i0 /. r is Element of the carrier of G
i0 . r is Element of the carrier of G
r * (m - X) is Element of the carrier of G
K458( the Mult of G,r,(m - X)) is set
(r * (m - X)) + X is Element of the carrier of G
K462( the carrier of G, the addF of G,(r * (m - X)),X) is Element of the carrier of G
(i0 | [.0,1.]) `| ].0,1.[ is Relation-like REAL -defined the carrier of G -valued Function-like Element of K6(K7(REAL, the carrier of G))
r is V11() real ext-real Element of REAL
((i0 | [.0,1.]) `| ].0,1.[) . r is set
K7(REAL, the carrier of F) is non trivial Relation-like V49() set
K6(K7(REAL, the carrier of F)) is non trivial V49() set
f * (i0 | [.0,1.]) is Relation-like REAL -defined the carrier of F -valued Function-like Element of K6(K7(REAL, the carrier of F))
rng (i0 | [.0,1.]) is Element of K6( the carrier of G)
s is set
y1 is set
(i0 | [.0,1.]) . y1 is set
i0 . y1 is set
PP is V11() real ext-real Element of REAL
DD is V11() real ext-real Element of REAL
PP * (m - X) is Element of the carrier of G
K458( the Mult of G,PP,(m - X)) is set
X + (PP * (m - X)) is Element of the carrier of G
K462( the carrier of G, the addF of G,X,(PP * (m - X))) is Element of the carrier of G
1 - PP is V11() real ext-real Element of REAL
K38(PP) is V11() real ext-real set
K36(1,K38(PP)) is V11() real ext-real set
(1 - PP) * X is Element of the carrier of G
K458( the Mult of G,(1 - PP),X) is set
PP * m is Element of the carrier of G
K458( the Mult of G,PP,m) is set
((1 - PP) * X) + (PP * m) is Element of the carrier of G
K462( the carrier of G, the addF of G,((1 - PP) * X),(PP * m)) is Element of the carrier of G
{ (((1 - b₁) * X) + (b₁ * m)) where b₁ is V11() real ext-real Element of REAL : ( 0 <= b₁ & b₁ <= 1 ) } is set
r is Relation-like REAL -defined the carrier of F -valued Function-like Element of K6(K7(REAL, the carrier of F))
dom r is complex-membered ext-real-membered real-membered Element of K6(REAL)
s is V11() real ext-real set
r /. s is Element of the carrier of F
s * (m - X) is Element of the carrier of G
K458( the Mult of G,s,(m - X)) is set
X + (s * (m - X)) is Element of the carrier of G
K462( the carrier of G, the addF of G,X,(s * (m - X))) is Element of the carrier of G
f /. (X + (s * (m - X))) is Element of the carrier of F
r . s is set
(i0 | [.0,1.]) . s is set
f . ((i0 | [.0,1.]) . s) is set
i0 . s is set
s is V11() real ext-real set
(i0 | [.0,1.]) . s is set
(i0 | [.0,1.]) /. s is Element of the carrier of G
r | [.0,1.] is Relation-like REAL -defined [.0,1.] -defined REAL -defined the carrier of F -valued Function-like Element of K6(K7(REAL, the carrier of F))
s is V11() real ext-real Element of REAL
((i0 | [.0,1.]) `| ].0,1.[) . s is set
diff ((i0 | [.0,1.]),s) is Element of the carrier of G
(i0 | [.0,1.]) . s is set
(i0 | [.0,1.]) /. s is Element of the carrier of G
i0 . s is Element of the carrier of G
s * (m - X) is Element of the carrier of G
K458( the Mult of G,s,(m - X)) is set
X + (s * (m - X)) is Element of the carrier of G
K462( the carrier of G, the addF of G,X,(s * (m - X))) is Element of the carrier of G
y1 is V11() real ext-real Element of REAL
diff (r,s) is Element of the carrier of F
diff (f,(X + (s * (m - X)))) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F))
the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F)) is non empty set
(diff (f,(X + (s * (m - X))))) . (m - X) is Element of the carrier of F
s is V11() real ext-real Element of REAL
s 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))
s | [.0,1.] is Relation-like REAL -defined [.0,1.] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
PP is V11() real ext-real set
s . PP is V11() real ext-real Element of REAL
(i * ||.(m - X).||) * PP is V11() real ext-real Element of REAL
((i * ||.(m - X).||) * PP) + 0 is V11() real ext-real Element of REAL
dom s is non empty complex-membered ext-real-membered real-membered Element of K6(REAL)
dom (s | [.0,1.]) is complex-membered ext-real-membered real-membered Element of K6(REAL)
(s | [.0,1.]) | [.0,1.] is Relation-like REAL -defined [.0,1.] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
PP is V11() real ext-real Element of REAL
(s | [.0,1.]) . PP is V11() real ext-real Element of REAL
s . PP is V11() real ext-real Element of REAL
(i * ||.(m - X).||) * PP is V11() real ext-real Element of REAL
((i * ||.(m - X).||) * PP) + 0 is V11() real ext-real Element of REAL
(s | [.0,1.]) `| ].0,1.[ is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
PP is V11() real ext-real Element of REAL
((s | [.0,1.]) `| ].0,1.[) . PP is V11() real ext-real Element of REAL
PP is V11() real ext-real set
diff (r,PP) is Element of the carrier of F
||.(diff (r,PP)).|| is V11() real ext-real non negative Element of REAL
the normF of F . (diff (r,PP)) is V11() real ext-real Element of REAL
diff ((s | [.0,1.]),PP) is V11() real ext-real Element of REAL
PP * (m - X) is Element of the carrier of G
K458( the Mult of G,PP,(m - X)) is set
X + (PP * (m - X)) is Element of the carrier of G
K462( the carrier of G, the addF of G,X,(PP * (m - X))) is Element of the carrier of G
diff (f,(X + (PP * (m - X)))) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F))
(diff (f,(X + (PP * (m - X))))) . (m - X) is Element of the carrier of F
||.((diff (f,(X + (PP * (m - X))))) . (m - X)).|| is V11() real ext-real non negative Element of REAL
the normF of F . ((diff (f,(X + (PP * (m - X))))) . (m - X)) is V11() real ext-real Element of REAL
DD is non empty Relation-like the carrier of G -defined the carrier of F -valued Function-like total quasi_total V153(G,F) V154(G,F) Lipschitzian Element of K6(K7( the carrier of G, the carrier of F))
DD . (m - X) is Element of the carrier of F
||.(DD . (m - X)).|| is V11() real ext-real non negative Element of REAL
the normF of F . (DD . (m - X)) is V11() real ext-real Element of REAL
||.(diff (f,(X + (PP * (m - X))))).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators (G,F)) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F)) -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 (G,F)),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F)),REAL)) is non trivial V49() set
the normF of (R_NormSpace_of_BoundedLinearOperators (G,F)) . (diff (f,(X + (PP * (m - X))))) is V11() real ext-real Element of REAL
||.(diff (f,(X + (PP * (m - X))))).|| * ||.(m - X).|| is V11() real ext-real non negative Element of REAL
a is V11() real ext-real Element of REAL
((s | [.0,1.]) `| ].0,1.[) . PP is V11() real ext-real Element of REAL
r /. 1 is Element of the carrier of F
r /. 0 is Element of the carrier of F
(r /. 1) - (r /. 0) is Element of the carrier of F
- (r /. 0) is Element of the carrier of F
(r /. 1) + (- (r /. 0)) is Element of the carrier of F
K462( the carrier of F, the addF of F,(r /. 1),(- (r /. 0))) is Element of the carrier of F
||.((r /. 1) - (r /. 0)).|| is V11() real ext-real non negative Element of REAL
the normF of F . ((r /. 1) - (r /. 0)) is V11() real ext-real Element of REAL
(s | [.0,1.]) /. 1 is V11() real ext-real Element of REAL
(s | [.0,1.]) /. 0 is V11() real ext-real Element of REAL
((s | [.0,1.]) /. 1) - ((s | [.0,1.]) /. 0) is V11() real ext-real Element of REAL
K38(((s | [.0,1.]) /. 0)) is V11() real ext-real set
K36(((s | [.0,1.]) /. 1),K38(((s | [.0,1.]) /. 0))) is V11() real ext-real set
(s | [.0,1.]) . 1 is V11() real ext-real Element of REAL
s . 1 is V11() real ext-real Element of REAL
(i * ||.(m - X).||) * 1 is V11() real ext-real Element of REAL
(s | [.0,1.]) . 0 is V11() real ext-real Element of REAL
s . 0 is V11() real ext-real Element of REAL
(i * ||.(m - X).||) * 0 is V11() real ext-real Element of REAL
1 * (m - X) is Element of the carrier of G
K458( the Mult of G,1,(m - X)) is set
X + (1 * (m - X)) is Element of the carrier of G
K462( the carrier of G, the addF of G,X,(1 * (m - X))) is Element of the carrier of G
f /. (X + (1 * (m - X))) is Element of the carrier of F
X + (m - X) is Element of the carrier of G
K462( the carrier of G, the addF of G,X,(m - X)) is Element of the carrier of G
f /. (X + (m - X)) is Element of the carrier of F
X - X is Element of the carrier of G
X + (- X) is Element of the carrier of G
K462( the carrier of G, the addF of G,X,(- X)) is Element of the carrier of G
m - (X - X) is Element of the carrier of G
- (X - X) is Element of the carrier of G
m + (- (X - X)) is Element of the carrier of G
K462( the carrier of G, the addF of G,m,(- (X - X))) is Element of the carrier of G
f /. (m - (X - X)) is Element of the carrier of F
0. G is V68(G) Element of the carrier of G
the ZeroF of G is Element of the carrier of G
m - (0. G) is Element of the carrier of G
- (0. G) is Element of the carrier of G
m + (- (0. G)) is Element of the carrier of G
K462( the carrier of G, the addF of G,m,(- (0. G))) is Element of the carrier of G
f /. (m - (0. G)) is Element of the carrier of F
0 * (m - X) is Element of the carrier of G
K458( the Mult of G,0,(m - X)) is set
X + (0 * (m - X)) is Element of the carrier of G
K462( the carrier of G, the addF of G,X,(0 * (m - X))) is Element of the carrier of G
f /. (X + (0 * (m - X))) is Element of the carrier of F
X + (0. G) is Element of the carrier of G
K462( the carrier of G, the addF of G,X,(0. G)) is Element of the carrier of G
f /. (X + (0. G)) is Element of the carrier of F
G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of G is non empty non trivial set
F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of F is non empty non trivial set
K7( the carrier of G, the carrier of F) is Relation-like set
K6(K7( the carrier of G, the carrier of F)) is set
R_NormSpace_of_BoundedLinearOperators (G,F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators (G,F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators (G,F)))
R_VectorSpace_of_LinearOperators (G,F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators (G,F) is non empty functional Element of K6( the carrier of K266( the carrier of G,F))
K266( the carrier of G,F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of G, the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of G, the carrier of F
FuncZero ( the carrier of G,F) is Relation-like the carrier of G -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of G, the carrier of F)
0. F is V68(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
the carrier of G --> (0. F) is non empty Relation-like the carrier of G -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of G, the carrier of F))
FuncAdd ( the carrier of G,F) is Relation-like K7((Funcs ( the carrier of G, the carrier of F)),(Funcs ( the carrier of G, the carrier of F))) -defined Funcs ( the carrier of G, the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of G, the carrier of F)),(Funcs ( the carrier of G, the carrier of F))),(Funcs ( the carrier of G, the carrier of F))))
K7((Funcs ( the carrier of G, the carrier of F)),(Funcs ( the carrier of G, the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of G, the carrier of F)),(Funcs ( the carrier of G, the carrier of F))),(Funcs ( the carrier of G, the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of G, the carrier of F)),(Funcs ( the carrier of G, the carrier of F))),(Funcs ( the carrier of G, the carrier of F)))) is set
FuncExtMult ( the carrier of G,F) is Relation-like K7(REAL,(Funcs ( the carrier of G, the carrier of F))) -defined Funcs ( the carrier of G, the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of G, the carrier of F))),(Funcs ( the carrier of G, the carrier of F))))
K7(REAL,(Funcs ( the carrier of G, the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of G, the carrier of F))),(Funcs ( the carrier of G, the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of G, the carrier of F))),(Funcs ( the carrier of G, the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of G, the carrier of F)),(FuncZero ( the carrier of G,F)),(FuncAdd ( the carrier of G,F)),(FuncExtMult ( the carrier of G,F)) #) is strict RLSStruct
the carrier of K266( the carrier of G,F) is non empty set
K6( the carrier of K266( the carrier of G,F)) is set
K215(K266( the carrier of G,F),(LinearOperators (G,F))) is Relation-like Function-like Element of LinearOperators (G,F)
K213(K266( the carrier of G,F),(LinearOperators (G,F))) is Relation-like K7((LinearOperators (G,F)),(LinearOperators (G,F))) -defined LinearOperators (G,F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators (G,F)),(LinearOperators (G,F))),(LinearOperators (G,F))))
K7((LinearOperators (G,F)),(LinearOperators (G,F))) is Relation-like set
K7(K7((LinearOperators (G,F)),(LinearOperators (G,F))),(LinearOperators (G,F))) is Relation-like set
K6(K7(K7((LinearOperators (G,F)),(LinearOperators (G,F))),(LinearOperators (G,F)))) is set
K214(K266( the carrier of G,F),(LinearOperators (G,F))) is Relation-like K7(REAL,(LinearOperators (G,F))) -defined LinearOperators (G,F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators (G,F))),(LinearOperators (G,F))))
K7(REAL,(LinearOperators (G,F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators (G,F))),(LinearOperators (G,F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators (G,F))),(LinearOperators (G,F)))) is non trivial V49() set
RLSStruct(# (LinearOperators (G,F)),K215(K266( the carrier of G,F),(LinearOperators (G,F))),K213(K266( the carrier of G,F),(LinearOperators (G,F))),K214(K266( the carrier of G,F),(LinearOperators (G,F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators (G,F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators (G,F))) is set
K215((R_VectorSpace_of_LinearOperators (G,F)),(BoundedLinearOperators (G,F))) is Element of BoundedLinearOperators (G,F)
K213((R_VectorSpace_of_LinearOperators (G,F)),(BoundedLinearOperators (G,F))) is Relation-like K7((BoundedLinearOperators (G,F)),(BoundedLinearOperators (G,F))) -defined BoundedLinearOperators (G,F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators (G,F)),(BoundedLinearOperators (G,F))),(BoundedLinearOperators (G,F))))
K7((BoundedLinearOperators (G,F)),(BoundedLinearOperators (G,F))) is Relation-like set
K7(K7((BoundedLinearOperators (G,F)),(BoundedLinearOperators (G,F))),(BoundedLinearOperators (G,F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators (G,F)),(BoundedLinearOperators (G,F))),(BoundedLinearOperators (G,F)))) is set
K214((R_VectorSpace_of_LinearOperators (G,F)),(BoundedLinearOperators (G,F))) is Relation-like K7(REAL,(BoundedLinearOperators (G,F))) -defined BoundedLinearOperators (G,F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators (G,F))),(BoundedLinearOperators (G,F))))
K7(REAL,(BoundedLinearOperators (G,F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators (G,F))),(BoundedLinearOperators (G,F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators (G,F))),(BoundedLinearOperators (G,F)))) is non trivial V49() set
BoundedLinearOperatorsNorm (G,F) is non empty Relation-like BoundedLinearOperators (G,F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators (G,F)),REAL))
K7((BoundedLinearOperators (G,F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators (G,F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators (G,F)),K215((R_VectorSpace_of_LinearOperators (G,F)),(BoundedLinearOperators (G,F))),K213((R_VectorSpace_of_LinearOperators (G,F)),(BoundedLinearOperators (G,F))),K214((R_VectorSpace_of_LinearOperators (G,F)),(BoundedLinearOperators (G,F))),(BoundedLinearOperatorsNorm (G,F)) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F)) is non empty set
f is Relation-like the carrier of G -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of G, the carrier of F))
dom f is Element of K6( the carrier of G)
K6( the carrier of G) is set
X is Element of the carrier of G
m is Element of the carrier of G
LSeg (X,m) is Element of K6( the carrier of G)
(G,X,m) is Element of K6( the carrier of G)
m - X is Element of the carrier of G
- X is Element of the carrier of G
m + (- X) is Element of the carrier of G
the addF of G is Relation-like K7( the carrier of G, the carrier of G) -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of G, the carrier of G), the carrier of G))
K7( the carrier of G, the carrier of G) is Relation-like set
K7(K7( the carrier of G, the carrier of G), the carrier of G) is Relation-like set
K6(K7(K7( the carrier of G, the carrier of G), the carrier of G)) is set
K462( the carrier of G, the addF of G,m,(- X)) is Element of the carrier of G
{ (X + (b₁ * (m - X))) where b₁ is V11() real ext-real Element of REAL : ( not b₁ <= 0 & not 1 <= b₁ ) } is set
f /. m is Element of the carrier of F
f /. X is Element of the carrier of F
(f /. m) - (f /. X) is Element of the carrier of F
- (f /. X) is Element of the carrier of F
(f /. m) + (- (f /. X)) is Element of the carrier of F
the addF of F is Relation-like K7( the carrier of F, the carrier of F) -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of F, the carrier of F), the carrier of F))
K7( the carrier of F, the carrier of F) is Relation-like set
K7(K7( the carrier of F, the carrier of F), the carrier of F) is Relation-like set
K6(K7(K7( the carrier of F, the carrier of F), the carrier of F)) is set
K462( the carrier of F, the addF of F,(f /. m),(- (f /. X))) is Element of the carrier of F
||.(m - X).|| is V11() real ext-real non negative Element of REAL
the normF of G is non empty Relation-like the carrier of G -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of G,REAL))
K7( the carrier of G,REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of G,REAL)) is non trivial V49() set
the normF of G . (m - X) is V11() real ext-real Element of REAL
i is V11() real ext-real Element of REAL
i * ||.(m - X).|| is V11() real ext-real Element of REAL
i0 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F))
i0 . (m - X) is Element of the carrier of F
((f /. m) - (f /. X)) - (i0 . (m - X)) is Element of the carrier of F
- (i0 . (m - X)) is Element of the carrier of F
((f /. m) - (f /. X)) + (- (i0 . (m - X))) is Element of the carrier of F
K462( the carrier of F, the addF of F,((f /. m) - (f /. X)),(- (i0 . (m - X)))) is Element of the carrier of F
||.(((f /. m) - (f /. X)) - (i0 . (m - X))).|| is V11() real ext-real non negative Element of REAL
the normF of F is non empty Relation-like the carrier of F -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of F,REAL))
K7( the carrier of F,REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of F,REAL)) is non trivial V49() set
the normF of F . (((f /. m) - (f /. X)) - (i0 . (m - X))) is V11() real ext-real Element of REAL
r is non empty Relation-like the carrier of G -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7( the carrier of G, the carrier of F))
dom r is non empty Element of K6( the carrier of G)
s is Element of the carrier of G
y1 is Element of the carrier of G
- 1 is non empty V11() real ext-real non positive negative V132() V160() Element of INT
r /. s is Element of the carrier of F
r . s is Element of the carrier of F
s - X is Element of the carrier of G
s + (- X) is Element of the carrier of G
K462( the carrier of G, the addF of G,s,(- X)) is Element of the carrier of G
i0 . (s - X) is Element of the carrier of F
r /. y1 is Element of the carrier of F
r . y1 is Element of the carrier of F
y1 - X is Element of the carrier of G
y1 + (- X) is Element of the carrier of G
K462( the carrier of G, the addF of G,y1,(- X)) is Element of the carrier of G
i0 . (y1 - X) is Element of the carrier of F
(r /. s) - (r /. y1) is Element of the carrier of F
- (r /. y1) is Element of the carrier of F
(r /. s) + (- (r /. y1)) is Element of the carrier of F
K462( the carrier of F, the addF of F,(r /. s),(- (r /. y1))) is Element of the carrier of F
||.((r /. s) - (r /. y1)).|| is V11() real ext-real non negative Element of REAL
the normF of F . ((r /. s) - (r /. y1)) is V11() real ext-real Element of REAL
y0 is non empty Relation-like the carrier of G -defined the carrier of F -valued Function-like total quasi_total V153(G,F) V154(G,F) Lipschitzian Element of K6(K7( the carrier of G, the carrier of F))
y0 . (s - X) is Element of the carrier of F
y0 . (y1 - X) is Element of the carrier of F
(- 1) * (y0 . (y1 - X)) is Element of the carrier of F
the Mult of F is Relation-like K7(REAL, the carrier of F) -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of F), the carrier of F))
K7(REAL, the carrier of F) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of F), the carrier of F) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of F), the carrier of F)) is non trivial V49() set
K458( the Mult of F,(- 1),(y0 . (y1 - X))) is set
(y0 . (s - X)) + ((- 1) * (y0 . (y1 - X))) is Element of the carrier of F
K462( the carrier of F, the addF of F,(y0 . (s - X)),((- 1) * (y0 . (y1 - X)))) is Element of the carrier of F
||.((y0 . (s - X)) + ((- 1) * (y0 . (y1 - X)))).|| is V11() real ext-real non negative Element of REAL
the normF of F . ((y0 . (s - X)) + ((- 1) * (y0 . (y1 - X)))) is V11() real ext-real Element of REAL
(- 1) * (y1 - X) is Element of the carrier of G
the Mult of G is Relation-like K7(REAL, the carrier of G) -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of G), the carrier of G))
K7(REAL, the carrier of G) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of G), the carrier of G) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of G), the carrier of G)) is non trivial V49() set
K458( the Mult of G,(- 1),(y1 - X)) is set
y0 . ((- 1) * (y1 - X)) is Element of the carrier of F
(y0 . (s - X)) + (y0 . ((- 1) * (y1 - X))) is Element of the carrier of F
K462( the carrier of F, the addF of F,(y0 . (s - X)),(y0 . ((- 1) * (y1 - X)))) is Element of the carrier of F
||.((y0 . (s - X)) + (y0 . ((- 1) * (y1 - X)))).|| is V11() real ext-real non negative Element of REAL
the normF of F . ((y0 . (s - X)) + (y0 . ((- 1) * (y1 - X)))) is V11() real ext-real Element of REAL
(s - X) + ((- 1) * (y1 - X)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(s - X),((- 1) * (y1 - X))) is Element of the carrier of G
y0 . ((s - X) + ((- 1) * (y1 - X))) is Element of the carrier of F
||.(y0 . ((s - X) + ((- 1) * (y1 - X)))).|| is V11() real ext-real non negative Element of REAL
the normF of F . (y0 . ((s - X) + ((- 1) * (y1 - X)))) is V11() real ext-real Element of REAL
(s - X) - (y1 - X) is Element of the carrier of G
- (y1 - X) is Element of the carrier of G
(s - X) + (- (y1 - X)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(s - X),(- (y1 - X))) is Element of the carrier of G
y0 . ((s - X) - (y1 - X)) is Element of the carrier of F
||.(y0 . ((s - X) - (y1 - X))).|| is V11() real ext-real non negative Element of REAL
the normF of F . (y0 . ((s - X) - (y1 - X))) is V11() real ext-real Element of REAL
(y1 - X) + X is Element of the carrier of G
K462( the carrier of G, the addF of G,(y1 - X),X) is Element of the carrier of G
s - ((y1 - X) + X) is Element of the carrier of G
- ((y1 - X) + X) is Element of the carrier of G
s + (- ((y1 - X) + X)) is Element of the carrier of G
K462( the carrier of G, the addF of G,s,(- ((y1 - X) + X))) is Element of the carrier of G
y0 . (s - ((y1 - X) + X)) is Element of the carrier of F
||.(y0 . (s - ((y1 - X) + X))).|| is V11() real ext-real non negative Element of REAL
the normF of F . (y0 . (s - ((y1 - X) + X))) is V11() real ext-real Element of REAL
X - X is Element of the carrier of G
X + (- X) is Element of the carrier of G
K462( the carrier of G, the addF of G,X,(- X)) is Element of the carrier of G
y1 - (X - X) is Element of the carrier of G
- (X - X) is Element of the carrier of G
y1 + (- (X - X)) is Element of the carrier of G
K462( the carrier of G, the addF of G,y1,(- (X - X))) is Element of the carrier of G
s - (y1 - (X - X)) is Element of the carrier of G
- (y1 - (X - X)) is Element of the carrier of G
s + (- (y1 - (X - X))) is Element of the carrier of G
K462( the carrier of G, the addF of G,s,(- (y1 - (X - X)))) is Element of the carrier of G
y0 . (s - (y1 - (X - X))) is Element of the carrier of F
||.(y0 . (s - (y1 - (X - X)))).|| is V11() real ext-real non negative Element of REAL
the normF of F . (y0 . (s - (y1 - (X - X)))) is V11() real ext-real Element of REAL
0. G is V68(G) Element of the carrier of G
the ZeroF of G is Element of the carrier of G
y1 - (0. G) is Element of the carrier of G
- (0. G) is Element of the carrier of G
y1 + (- (0. G)) is Element of the carrier of G
K462( the carrier of G, the addF of G,y1,(- (0. G))) is Element of the carrier of G
s - (y1 - (0. G)) is Element of the carrier of G
- (y1 - (0. G)) is Element of the carrier of G
s + (- (y1 - (0. G))) is Element of the carrier of G
K462( the carrier of G, the addF of G,s,(- (y1 - (0. G)))) is Element of the carrier of G
y0 . (s - (y1 - (0. G))) is Element of the carrier of F
||.(y0 . (s - (y1 - (0. G)))).|| is V11() real ext-real non negative Element of REAL
the normF of F . (y0 . (s - (y1 - (0. G)))) is V11() real ext-real Element of REAL
s - y1 is Element of the carrier of G
- y1 is Element of the carrier of G
s + (- y1) is Element of the carrier of G
K462( the carrier of G, the addF of G,s,(- y1)) is Element of the carrier of G
y0 . (s - y1) is Element of the carrier of F
||.(y0 . (s - y1)).|| is V11() real ext-real non negative Element of REAL
the normF of F . (y0 . (s - y1)) is V11() real ext-real Element of REAL
||.i0.|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators (G,F)) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F)) -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 (G,F)),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F)),REAL)) is non trivial V49() set
the normF of (R_NormSpace_of_BoundedLinearOperators (G,F)) . i0 is V11() real ext-real Element of REAL
||.(s - y1).|| is V11() real ext-real non negative Element of REAL
the normF of G . (s - y1) is V11() real ext-real Element of REAL
||.i0.|| * ||.(s - y1).|| is V11() real ext-real non negative Element of REAL
1 + ||.i0.|| is non empty V11() real ext-real positive non negative Element of REAL
0 + ||.i0.|| is V11() real ext-real non negative Element of REAL
||.i0.|| + 1 is non empty V11() real ext-real positive non negative Element of REAL
(||.i0.|| + 1) * ||.(s - y1).|| is V11() real ext-real non negative Element of REAL
s is Relation-like the carrier of G -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of G, the carrier of F))
dom s is Element of K6( the carrier of G)
K7(NAT, the carrier of G) is non trivial Relation-like V49() set
K6(K7(NAT, the carrier of G)) is non trivial V49() set
y1 is non empty Relation-like NAT -defined the carrier of G -valued Function-like total quasi_total non-zero 0. G -convergent Element of K6(K7(NAT, the carrier of G))
PP is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
y1 . PP is Element of the carrier of G
s /. (y1 . PP) is Element of the carrier of F
s . (y1 . PP) is set
rng y1 is non empty Element of K6( the carrier of G)
s /* y1 is non empty Relation-like NAT -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of F))
K7(NAT, the carrier of F) is non trivial Relation-like V49() set
K6(K7(NAT, the carrier of F)) is non trivial V49() set
||.y1.|| is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
||.y1.|| " is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(||.y1.|| ") (#) (s /* y1) is non empty Relation-like NAT -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of F))
((||.y1.|| ") (#) (s /* y1)) . PP is Element of the carrier of F
(s /* y1) . PP is Element of the carrier of F
(||.y1.|| ") . PP is V11() real ext-real Element of REAL
((||.y1.|| ") . PP) * ((s /* y1) . PP) is Element of the carrier of F
K458( the Mult of F,((||.y1.|| ") . PP),((s /* y1) . PP)) is set
((||.y1.|| ") . PP) * (s /. (y1 . PP)) is Element of the carrier of F
K458( the Mult of F,((||.y1.|| ") . PP),(s /. (y1 . PP))) is set
((||.y1.|| ") (#) (s /* y1)) . 0 is Element of the carrier of F
lim ((||.y1.|| ") (#) (s /* y1)) is Element of the carrier of F
PP is Element of the carrier of G
the Neighbourhood of PP is Neighbourhood of PP
r /. PP is Element of the carrier of F
r . PP is Element of the carrier of F
y1 is Relation-like the carrier of G -defined the carrier of F -valued Function-like RestFunc-like Element of K6(K7( the carrier of G, the carrier of F))
a is Element of the carrier of G
r /. a is Element of the carrier of F
r . a is Element of the carrier of F
(r /. a) - (r /. PP) is Element of the carrier of F
- (r /. PP) is Element of the carrier of F
(r /. a) + (- (r /. PP)) is Element of the carrier of F
K462( the carrier of F, the addF of F,(r /. a),(- (r /. PP))) is Element of the carrier of F
a - PP is Element of the carrier of G
- PP is Element of the carrier of G
a + (- PP) is Element of the carrier of G
K462( the carrier of G, the addF of G,a,(- PP)) is Element of the carrier of G
i0 . (a - PP) is Element of the carrier of F
y1 /. (a - PP) is Element of the carrier of F
(i0 . (a - PP)) + (y1 /. (a - PP)) is Element of the carrier of F
K462( the carrier of F, the addF of F,(i0 . (a - PP)),(y1 /. (a - PP))) is Element of the carrier of F
y1 . (a - PP) is set
a - X is Element of the carrier of G
a + (- X) is Element of the carrier of G
K462( the carrier of G, the addF of G,a,(- X)) is Element of the carrier of G
i0 . (a - X) is Element of the carrier of F
(i0 . (a - X)) - (r . PP) is Element of the carrier of F
- (r . PP) is Element of the carrier of F
(i0 . (a - X)) + (- (r . PP)) is Element of the carrier of F
K462( the carrier of F, the addF of F,(i0 . (a - X)),(- (r . PP))) is Element of the carrier of F
PP - X is Element of the carrier of G
PP + (- X) is Element of the carrier of G
K462( the carrier of G, the addF of G,PP,(- X)) is Element of the carrier of G
i0 . (PP - X) is Element of the carrier of F
(i0 . (a - X)) - (i0 . (PP - X)) is Element of the carrier of F
- (i0 . (PP - X)) is Element of the carrier of F
(i0 . (a - X)) + (- (i0 . (PP - X))) is Element of the carrier of F
K462( the carrier of F, the addF of F,(i0 . (a - X)),(- (i0 . (PP - X)))) is Element of the carrier of F
y0 . (a - X) is Element of the carrier of F
y0 . (PP - X) is Element of the carrier of F
(- 1) * (y0 . (PP - X)) is Element of the carrier of F
K458( the Mult of F,(- 1),(y0 . (PP - X))) is set
(y0 . (a - X)) + ((- 1) * (y0 . (PP - X))) is Element of the carrier of F
K462( the carrier of F, the addF of F,(y0 . (a - X)),((- 1) * (y0 . (PP - X)))) is Element of the carrier of F
(- 1) * (PP - X) is Element of the carrier of G
K458( the Mult of G,(- 1),(PP - X)) is set
y0 . ((- 1) * (PP - X)) is Element of the carrier of F
(y0 . (a - X)) + (y0 . ((- 1) * (PP - X))) is Element of the carrier of F
K462( the carrier of F, the addF of F,(y0 . (a - X)),(y0 . ((- 1) * (PP - X)))) is Element of the carrier of F
(a - X) + ((- 1) * (PP - X)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(a - X),((- 1) * (PP - X))) is Element of the carrier of G
y0 . ((a - X) + ((- 1) * (PP - X))) is Element of the carrier of F
(a - X) - (PP - X) is Element of the carrier of G
- (PP - X) is Element of the carrier of G
(a - X) + (- (PP - X)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(a - X),(- (PP - X))) is Element of the carrier of G
y0 . ((a - X) - (PP - X)) is Element of the carrier of F
(PP - X) + X is Element of the carrier of G
K462( the carrier of G, the addF of G,(PP - X),X) is Element of the carrier of G
a - ((PP - X) + X) is Element of the carrier of G
- ((PP - X) + X) is Element of the carrier of G
a + (- ((PP - X) + X)) is Element of the carrier of G
K462( the carrier of G, the addF of G,a,(- ((PP - X) + X))) is Element of the carrier of G
y0 . (a - ((PP - X) + X)) is Element of the carrier of F
PP - (X - X) is Element of the carrier of G
PP + (- (X - X)) is Element of the carrier of G
K462( the carrier of G, the addF of G,PP,(- (X - X))) is Element of the carrier of G
a - (PP - (X - X)) is Element of the carrier of G
- (PP - (X - X)) is Element of the carrier of G
a + (- (PP - (X - X))) is Element of the carrier of G
K462( the carrier of G, the addF of G,a,(- (PP - (X - X)))) is Element of the carrier of G
y0 . (a - (PP - (X - X))) is Element of the carrier of F
PP - (0. G) is Element of the carrier of G
PP + (- (0. G)) is Element of the carrier of G
K462( the carrier of G, the addF of G,PP,(- (0. G))) is Element of the carrier of G
a - (PP - (0. G)) is Element of the carrier of G
- (PP - (0. G)) is Element of the carrier of G
a + (- (PP - (0. G))) is Element of the carrier of G
K462( the carrier of G, the addF of G,a,(- (PP - (0. G)))) is Element of the carrier of G
y0 . (a - (PP - (0. G))) is Element of the carrier of F
y0 . (a - PP) is Element of the carrier of F
diff (r,PP) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F))
f - r is Relation-like the carrier of G -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of G, the carrier of F))
dom (f - r) is Element of K6( the carrier of G)
(dom f) /\ (dom r) is Element of K6( the carrier of G)
DD is Element of the carrier of G
(f - r) /. DD is Element of the carrier of F
f /. DD is Element of the carrier of F
DD - X is Element of the carrier of G
DD + (- X) is Element of the carrier of G
K462( the carrier of G, the addF of G,DD,(- X)) is Element of the carrier of G
i0 . (DD - X) is Element of the carrier of F
(f /. DD) - (i0 . (DD - X)) is Element of the carrier of F
- (i0 . (DD - X)) is Element of the carrier of F
(f /. DD) + (- (i0 . (DD - X))) is Element of the carrier of F
K462( the carrier of F, the addF of F,(f /. DD),(- (i0 . (DD - X)))) is Element of the carrier of F
r /. DD is Element of the carrier of F
r . DD is Element of the carrier of F
(f /. DD) - (r /. DD) is Element of the carrier of F
- (r /. DD) is Element of the carrier of F
(f /. DD) + (- (r /. DD)) is Element of the carrier of F
K462( the carrier of F, the addF of F,(f /. DD),(- (r /. DD))) is Element of the carrier of F
DD is Element of the carrier of G
r | (dom r) is Relation-like the carrier of G -defined dom r -defined the carrier of G -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of G, the carrier of F))
DD is Element of the carrier of G
DD is Element of the carrier of G
diff ((f - r),DD) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F))
||.(diff ((f - r),DD)).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators (G,F)) . (diff ((f - r),DD)) is V11() real ext-real Element of REAL
diff (r,DD) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F))
diff (f,DD) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F))
(diff (f,DD)) - i0 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F))
- i0 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F))
(diff (f,DD)) + (- i0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F))
the addF of (R_NormSpace_of_BoundedLinearOperators (G,F)) is Relation-like K7( the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F)), the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F))) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F)) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F)), the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F))), the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F))))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F)), the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F))) is Relation-like set
K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F)), the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F))), the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F))) is Relation-like set
K6(K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F)), the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F))), the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F)))) is set
K462( the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F)), the addF of (R_NormSpace_of_BoundedLinearOperators (G,F)),(diff (f,DD)),(- i0)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (G,F))
(f - r) /. m is Element of the carrier of F
(f - r) /. X is Element of the carrier of F
((f - r) /. m) - ((f - r) /. X) is Element of the carrier of F
- ((f - r) /. X) is Element of the carrier of F
((f - r) /. m) + (- ((f - r) /. X)) is Element of the carrier of F
K462( the carrier of F, the addF of F,((f - r) /. m),(- ((f - r) /. X))) is Element of the carrier of F
||.(((f - r) /. m) - ((f - r) /. X)).|| is V11() real ext-real non negative Element of REAL
the normF of F . (((f - r) /. m) - ((f - r) /. X)) is V11() real ext-real Element of REAL
i0 . (X - X) is Element of the carrier of F
(f /. X) - (i0 . (X - X)) is Element of the carrier of F
- (i0 . (X - X)) is Element of the carrier of F
(f /. X) + (- (i0 . (X - X))) is Element of the carrier of F
K462( the carrier of F, the addF of F,(f /. X),(- (i0 . (X - X)))) is Element of the carrier of F
y0 . (0. G) is Element of the carrier of F
(f /. X) - (y0 . (0. G)) is Element of the carrier of F
- (y0 . (0. G)) is Element of the carrier of F
(f /. X) + (- (y0 . (0. G))) is Element of the carrier of F
K462( the carrier of F, the addF of F,(f /. X),(- (y0 . (0. G)))) is Element of the carrier of F
0 * X is Element of the carrier of G
K458( the Mult of G,0,X) is set
y0 . (0 * X) is Element of the carrier of F
(f /. X) - (y0 . (0 * X)) is Element of the carrier of F
- (y0 . (0 * X)) is Element of the carrier of F
(f /. X) + (- (y0 . (0 * X))) is Element of the carrier of F
K462( the carrier of F, the addF of F,(f /. X),(- (y0 . (0 * X)))) is Element of the carrier of F
y0 . X is Element of the carrier of F
0 * (y0 . X) is Element of the carrier of F
K458( the Mult of F,0,(y0 . X)) is set
(f /. X) - (0 * (y0 . X)) is Element of the carrier of F
- (0 * (y0 . X)) is Element of the carrier of F
(f /. X) + (- (0 * (y0 . X))) is Element of the carrier of F
K462( the carrier of F, the addF of F,(f /. X),(- (0 * (y0 . X)))) is Element of the carrier of F
(f /. X) - (0. F) is Element of the carrier of F
- (0. F) is Element of the carrier of F
(f /. X) + (- (0. F)) is Element of the carrier of F
K462( the carrier of F, the addF of F,(f /. X),(- (0. F))) is Element of the carrier of F
(f /. m) - (i0 . (m - X)) is Element of the carrier of F
(f /. m) + (- (i0 . (m - X))) is Element of the carrier of F
K462( the carrier of F, the addF of F,(f /. m),(- (i0 . (m - X)))) is Element of the carrier of F
(i0 . (m - X)) + (f /. X) is Element of the carrier of F
K462( the carrier of F, the addF of F,(i0 . (m - X)),(f /. X)) is Element of the carrier of F
(f /. m) - ((i0 . (m - X)) + (f /. X)) is Element of the carrier of F
- ((i0 . (m - X)) + (f /. X)) is Element of the carrier of F
(f /. m) + (- ((i0 . (m - X)) + (f /. X))) is Element of the carrier of F
K462( the carrier of F, the addF of F,(f /. m),(- ((i0 . (m - X)) + (f /. X)))) is Element of the carrier of F
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
product G is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty set
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . F is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . F) is non empty set
K7( the carrier of (product G), the carrier of (G . F)) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . F))) is set
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
G . F is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . F) is non empty set
K7((product (carr G)), the carrier of (G . F)) is Relation-like set
K6(K7((product (carr G)), the carrier of (G . F))) is set
f is non empty Relation-like product (carr G) -defined the carrier of (G . F) -valued Function-like total quasi_total Element of K6(K7((product (carr G)), the carrier of (G . F)))
zeros G is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
addop G is Relation-like Function-like BinOps of carr G
[:(addop G):] is Relation-like K7((product (carr G)),(product (carr G))) -defined product (carr G) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((product (carr G)),(product (carr G))),(product (carr G))))
K7((product (carr G)),(product (carr G))) is Relation-like set
K7(K7((product (carr G)),(product (carr G))),(product (carr G))) is Relation-like set
K6(K7(K7((product (carr G)),(product (carr G))),(product (carr G)))) is set
multop G is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr G
[:(multop G):] is Relation-like K7(REAL,(product (carr G))) -defined product (carr G) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(product (carr G))),(product (carr G))))
K7(REAL,(product (carr G))) is non trivial Relation-like V49() set
K7(K7(REAL,(product (carr G))),(product (carr G))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(product (carr G))),(product (carr G)))) is non trivial V49() set
productnorm G is non empty Relation-like product (carr G) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((product (carr G)),REAL))
K7((product (carr G)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((product (carr G)),REAL)) is non trivial V49() set
NORMSTR(# (product (carr G)),(zeros G),[:(addop G):],[:(multop G):],(productnorm G) #) is strict NORMSTR
X is non empty Relation-like the carrier of (product G) -defined the carrier of (G . F) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . F)))
m is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
X . m is set
(G,m,F) is Element of the carrier of (G . F)
f is non empty Relation-like the carrier of (product G) -defined the carrier of (G . F) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . F)))
X is non empty Relation-like the carrier of (product G) -defined the carrier of (G . F) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . F)))
zeros G is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
addop G is Relation-like Function-like BinOps of carr G
[:(addop G):] is Relation-like K7((product (carr G)),(product (carr G))) -defined product (carr G) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((product (carr G)),(product (carr G))),(product (carr G))))
K7((product (carr G)),(product (carr G))) is Relation-like set
K7(K7((product (carr G)),(product (carr G))),(product (carr G))) is Relation-like set
K6(K7(K7((product (carr G)),(product (carr G))),(product (carr G)))) is set
multop G is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr G
[:(multop G):] is Relation-like K7(REAL,(product (carr G))) -defined product (carr G) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(product (carr G))),(product (carr G))))
K7(REAL,(product (carr G))) is non trivial Relation-like V49() set
K7(K7(REAL,(product (carr G))),(product (carr G))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(product (carr G))),(product (carr G)))) is non trivial V49() set
productnorm G is non empty Relation-like product (carr G) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((product (carr G)),REAL))
K7((product (carr G)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((product (carr G)),REAL)) is non trivial V49() set
NORMSTR(# (product (carr G)),(zeros G),[:(addop G):],[:(multop G):],(productnorm G) #) is strict NORMSTR
m is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f . m is Element of the carrier of (G . F)
i is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
(G,i,F) is Element of the carrier of (G . F)
G . F is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . F) is non empty set
X . m is Element of the carrier of (G . F)
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
product G is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty set
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . F is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . F) is non empty set
K7( the carrier of (G . F), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . F), the carrier of (product G))) is set
f is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
X is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
m is Element of the carrier of (G . F)
Replace (F,m,) is Relation-like Function-like set
m is non empty Relation-like the carrier of (G . F) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . F), the carrier of (product G)))
X is non empty Relation-like the carrier of (G . F) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . F), the carrier of (product G)))
m is non empty Relation-like the carrier of (G . F) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . F), the carrier of (product G)))
i is Element of the carrier of (G . F)
X . i is set
m . i is set
X . i is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
Replace (F,i,) is Relation-like Function-like set
m . i is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
F is set
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of F is non empty non trivial set
K7( the carrier of (product G), the carrier of F) is Relation-like set
K6(K7( the carrier of (product G), the carrier of F)) is set
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of F is non empty non trivial set
K7( the carrier of (product G), the carrier of F) is Relation-like set
K6(K7( the carrier of (product G), the carrier of F)) is set
f is set
(G,f) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
G . (G,f) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . (G,f)) is non empty non trivial set
m is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,(G,f),m) is non empty Relation-like the carrier of (G . (G,f)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,f)), the carrier of (product G)))
K7( the carrier of (G . (G,f)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,f)), the carrier of (product G))) is set
X is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of F))
X * (G,(G,f),m) is Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,f)), the carrier of F))
K7( the carrier of (G . (G,f)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,f)), the carrier of F)) is set
(G,(G,f)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,f)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,f))))
K7( the carrier of (product G), the carrier of (G . (G,f))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,f)))) is set
(G,(G,f)) . m is Element of the carrier of (G . (G,f))
diff ((X * (G,(G,f),m)),((G,(G,f)) . m)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F))
R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,f)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,f)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,f)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,f)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,f)),F))
K266( the carrier of (G . (G,f)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,f)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,f)), the carrier of F
FuncZero ( the carrier of (G . (G,f)),F) is Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,f)), the carrier of F)
0. F is V68(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
the carrier of (G . (G,f)) --> (0. F) is non empty Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,f)), the carrier of F))
FuncAdd ( the carrier of (G . (G,f)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,f)), the carrier of F)),(Funcs ( the carrier of (G . (G,f)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,f)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,f)), the carrier of F)),(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,f)), the carrier of F)),(Funcs ( the carrier of (G . (G,f)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,f)), the carrier of F)),(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,f)), the carrier of F)),(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,f)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,f)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,f)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,f)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,f)), the carrier of F)),(FuncZero ( the carrier of (G . (G,f)),F)),(FuncAdd ( the carrier of (G . (G,f)),F)),(FuncExtMult ( the carrier of (G . (G,f)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,f)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,f)),F)) is set
K215(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,f)),F)
K213(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))) is Relation-like K7((LinearOperators ((G . (G,f)),F)),(LinearOperators ((G . (G,f)),F))) -defined LinearOperators ((G . (G,f)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,f)),F)),(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F))))
K7((LinearOperators ((G . (G,f)),F)),(LinearOperators ((G . (G,f)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,f)),F)),(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,f)),F)),(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F)))) is set
K214(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,f)),F))) -defined LinearOperators ((G . (G,f)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F))))
K7(REAL,(LinearOperators ((G . (G,f)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,f)),F)),K215(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))),K213(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))),K214(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,f)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,f)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))) is Element of BoundedLinearOperators ((G . (G,f)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))) -defined BoundedLinearOperators ((G . (G,f)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F))))
K7((BoundedLinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,f)),F))) -defined BoundedLinearOperators ((G . (G,f)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,f)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,f)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,f)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,f)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,f)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,f)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,f)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperatorsNorm ((G . (G,f)),F)) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F)) is non empty set
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of F is non empty non trivial set
K7( the carrier of (product G), the carrier of F) is Relation-like set
K6(K7( the carrier of (product G), the carrier of F)) is set
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
0. (product G) is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like V68( product G) countable Element of the carrier of (product G)
the ZeroF of (product G) is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . F) is non empty non trivial set
(G,F,(0. (product G))) is non empty Relation-like the carrier of (G . F) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . F), the carrier of (product G)))
K7( the carrier of (G . F), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . F), the carrier of (product G))) is set
f is Element of the carrier of (G . F)
(G,F,(0. (product G))) . f is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.((G,F,(0. (product G))) . f).|| is V11() real ext-real non negative Element of REAL
the normF of (product G) is non empty Relation-like the carrier of (product G) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (product G),REAL))
K7( the carrier of (product G),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (product G),REAL)) is non trivial V49() set
the normF of (product G) . ((G,F,(0. (product G))) . f) is V11() real ext-real Element of REAL
||.f.|| is V11() real ext-real non negative Element of REAL
the normF of (G . F) is non empty Relation-like the carrier of (G . F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (G . F),REAL))
K7( the carrier of (G . F),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (G . F),REAL)) is non trivial V49() set
the normF of (G . F) . f is V11() real ext-real Element of REAL
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len G) is non empty V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= len G ) } is set
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
Replace (F,f,) is Relation-like Function-like set
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
Replace (m,f,) is Relation-like Function-like set
i0 is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
normsequence (G,i0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() len G -element FinSequence-like FinSubsequence-like countable Element of REAL (len G)
REAL (len G) is non empty functional FinSequence-membered FinSequenceSet of REAL
(len G) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
REAL * is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like V49() FinSequence-like FinSubsequence-like countable Element of REAL * : len b₁ = len G } is set
|.(normsequence (G,i0)).| is V11() real ext-real non negative Element of REAL
sqr (normsequence (G,i0)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
Sum (sqr (normsequence (G,i0))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (normsequence (G,i0)))) is V11() real ext-real Element of REAL
<*||.f.||*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing V49() 1 -element FinSequence-like FinSubsequence-like countable FinSequence of REAL
[1,||.f.||] is set
{1,||.f.||} is non empty V49() complex-membered ext-real-membered real-membered countable left_end right_end bounded_below bounded_above real-bounded set
{{1,||.f.||},{1}} is non empty V49() V53() countable set
{[1,||.f.||]} is non empty trivial Relation-like Function-like constant V49() 1 -element countable set
0* (len G) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() len G -element FinSequence-like FinSubsequence-like countable Element of REAL (len G)
(len G) |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() len G -element FinSequence-like FinSubsequence-like countable Element of (len G) -tuples_on REAL
(Seg (len G)) --> 0 is non empty Relation-like NAT -defined Seg (len G) -defined Seg (len G) -defined NAT -valued RAT -valued INT -valued {0} -valued Function-like constant total total quasi_total complex-valued ext-real-valued real-valued natural-valued V49() FinSequence-like FinSubsequence-like Function-yielding V134() Cardinal-yielding countable Element of K6(K7((Seg (len G)),{0}))
{0} is non empty trivial functional V49() V53() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered with_common_domain countable left_end right_end bounded_below bounded_above real-bounded set
K7((Seg (len G)),{0}) is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued V49() countable set
K6(K7((Seg (len G)),{0})) is V49() V53() countable set
len (normsequence (G,i0)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
len (0* (len G)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
m -' 1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
(0* (len G)) | (m -' 1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
Seg (m -' 1) is V49() m -' 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= m -' 1 ) } is set
(0* (len G)) | (Seg (m -' 1)) is Relation-like NAT -defined Seg (m -' 1) -defined NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSubsequence-like countable set
((0* (len G)) | (m -' 1)) ^ <*||.f.||*> is non empty Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable Element of REAL *
(0* (len G)) /^ m is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
(((0* (len G)) | (m -' 1)) ^ <*||.f.||*>) ^ ((0* (len G)) /^ m) is non empty Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable Element of REAL *
Replace ((0* (len G)),m,||.f.||) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
len ((((0* (len G)) | (m -' 1)) ^ <*||.f.||*>) ^ ((0* (len G)) /^ m)) is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
len (Replace ((0* (len G)),m,||.f.||)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
y1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
(normsequence (G,i0)) . y1 is V11() real ext-real Element of REAL
((((0* (len G)) | (m -' 1)) ^ <*||.f.||*>) ^ ((0* (len G)) /^ m)) . y1 is V11() real ext-real Element of REAL
PP is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
dom (carr G) is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
DD is Relation-like Function-like set
dom DD is set
dom (0. (product G)) is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
G . PP is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the normF of (G . PP) is non empty Relation-like the carrier of (G . PP) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (G . PP),REAL))
the carrier of (G . PP) is non empty non trivial set
K7( the carrier of (G . PP),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (G . PP),REAL)) is non trivial V49() set
(G,i0,PP) is Element of the carrier of (G . PP)
G . PP is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . PP) is non empty set
the normF of (G . PP) . (G,i0,PP) is V11() real ext-real Element of REAL
(Replace ((0* (len G)),m,||.f.||)) /. y1 is V11() real ext-real Element of REAL
(0. (product G)) . PP is set
0. (G . PP) is V68(G . PP) Element of the carrier of (G . PP)
the ZeroF of (G . PP) is Element of the carrier of (G . PP)
||.(0. (G . PP)).|| is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional complex-valued ext-real-valued real-valued natural-valued V49() V50() V53() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V134() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V167() Cardinal-yielding countable bounded_below bounded_above real-bounded V251() Element of REAL
the normF of (G . PP) . (0. (G . PP)) is V11() real ext-real Element of REAL
(Replace ((0* (len G)),m,||.f.||)) /. y1 is V11() real ext-real Element of REAL
(0* (len G)) /. y1 is V11() real ext-real Element of REAL
(Replace ((0* (len G)),m,||.f.||)) . y1 is V11() real ext-real Element of REAL
(0* (len G)) . y1 is V11() real ext-real Element of REAL
sqr ((0* (len G)) | (m -' 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
Sum (sqr ((0* (len G)) | (m -' 1))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr ((0* (len G)) | (m -' 1)))) is V11() real ext-real Element of REAL
0* (m -' 1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() m -' 1 -element FinSequence-like FinSubsequence-like countable Element of REAL (m -' 1)
REAL (m -' 1) is non empty functional FinSequence-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 V49() FinSequence-like FinSubsequence-like countable Element of REAL * : len b₁ = m -' 1 } is set
(m -' 1) |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() m -' 1 -element FinSequence-like FinSubsequence-like countable Element of (m -' 1) -tuples_on REAL
(Seg (m -' 1)) --> 0 is Relation-like NAT -defined Seg (m -' 1) -defined Seg (m -' 1) -defined NAT -valued RAT -valued INT -valued {0} -valued Function-like constant total total quasi_total complex-valued ext-real-valued real-valued natural-valued V49() FinSequence-like FinSubsequence-like Function-yielding V134() Cardinal-yielding countable Element of K6(K7((Seg (m -' 1)),{0}))
K7((Seg (m -' 1)),{0}) is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued V49() countable set
K6(K7((Seg (m -' 1)),{0})) is V49() V53() countable 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 complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
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
sqr ((0* (len G)) /^ m) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
Sum (sqr ((0* (len G)) /^ m)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr ((0* (len G)) /^ m))) is V11() real ext-real Element of REAL
(len G) -' m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
0* ((len G) -' m) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() (len G) -' m -element FinSequence-like FinSubsequence-like countable Element of REAL ((len G) -' m)
REAL ((len G) -' m) is non empty functional FinSequence-membered FinSequenceSet of REAL
((len G) -' 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 V49() FinSequence-like FinSubsequence-like countable Element of REAL * : len b₁ = (len G) -' m } is set
((len G) -' m) |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() (len G) -' m -element FinSequence-like FinSubsequence-like countable Element of ((len G) -' m) -tuples_on REAL
Seg ((len G) -' m) is V49() (len G) -' m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= (len G) -' m ) } is set
(Seg ((len G) -' m)) --> 0 is Relation-like NAT -defined Seg ((len G) -' m) -defined Seg ((len G) -' m) -defined NAT -valued RAT -valued INT -valued {0} -valued Function-like constant total total quasi_total complex-valued ext-real-valued real-valued natural-valued V49() FinSequence-like FinSubsequence-like Function-yielding V134() Cardinal-yielding countable Element of K6(K7((Seg ((len G) -' m)),{0}))
K7((Seg ((len G) -' m)),{0}) is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued V49() countable set
K6(K7((Seg ((len G) -' m)),{0})) is V49() V53() countable set
|.(0* ((len G) -' m)).| is V11() real ext-real non negative Element of REAL
sqr (0* ((len G) -' m)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
Sum (sqr (0* ((len G) -' m))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (0* ((len G) -' m)))) is V11() real ext-real Element of REAL
sqr ((((0* (len G)) | (m -' 1)) ^ <*||.f.||*>) ^ ((0* (len G)) /^ m)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
sqr (((0* (len G)) | (m -' 1)) ^ <*||.f.||*>) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
(sqr (((0* (len G)) | (m -' 1)) ^ <*||.f.||*>)) ^ (sqr ((0* (len G)) /^ m)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable Element of REAL *
sqr <*||.f.||*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
(sqr ((0* (len G)) | (m -' 1))) ^ (sqr <*||.f.||*>) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable Element of REAL *
((sqr ((0* (len G)) | (m -' 1))) ^ (sqr <*||.f.||*>)) ^ (sqr ((0* (len G)) /^ m)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable Element of REAL *
||.f.|| ^2 is V11() real ext-real Element of REAL
<*(||.f.|| ^2)*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing V49() 1 -element FinSequence-like FinSubsequence-like countable FinSequence of REAL
[1,(||.f.|| ^2)] is set
{1,(||.f.|| ^2)} is non empty V49() complex-membered ext-real-membered real-membered countable left_end right_end bounded_below bounded_above real-bounded set
{{1,(||.f.|| ^2)},{1}} is non empty V49() V53() countable set
{[1,(||.f.|| ^2)]} is non empty trivial Relation-like Function-like constant V49() 1 -element countable set
(sqr ((0* (len G)) | (m -' 1))) ^ <*(||.f.|| ^2)*> is non empty Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable Element of REAL *
((sqr ((0* (len G)) | (m -' 1))) ^ <*(||.f.|| ^2)*>) ^ (sqr ((0* (len G)) /^ m)) is non empty Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable Element of REAL *
Sum (sqr ((((0* (len G)) | (m -' 1)) ^ <*||.f.||*>) ^ ((0* (len G)) /^ m))) is V11() real ext-real Element of REAL
Sum ((sqr ((0* (len G)) | (m -' 1))) ^ <*(||.f.|| ^2)*>) is V11() real ext-real Element of REAL
(Sum ((sqr ((0* (len G)) | (m -' 1))) ^ <*(||.f.|| ^2)*>)) + (Sum (sqr ((0* (len G)) /^ m))) is V11() real ext-real Element of REAL
(Sum (sqr ((0* (len G)) | (m -' 1)))) + (||.f.|| ^2) is V11() real ext-real Element of REAL
((Sum (sqr ((0* (len G)) | (m -' 1)))) + (||.f.|| ^2)) + (Sum (sqr ((0* (len G)) /^ m))) is V11() real ext-real Element of REAL
|.||.f.||.| is V11() real ext-real Element of REAL
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
0. (product G) is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like V68( product G) countable Element of the carrier of (product G)
the ZeroF of (product G) is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . F) is non empty non trivial set
(G,F,(0. (product G))) is non empty Relation-like the carrier of (G . F) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . F), the carrier of (product G)))
K7( the carrier of (G . F), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . F), the carrier of (product G))) is set
(G,F) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . F) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . F)))
K7( the carrier of (product G), the carrier of (G . F)) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . F))) is set
f is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,F,f) is non empty Relation-like the carrier of (G . F) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . F), the carrier of (product G)))
(G,F) . f is Element of the carrier of (G . F)
X is Element of the carrier of (G . F)
(G,F,f) . X is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((G,F,f) . X) - f is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
- f is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((G,F,f) . X) + (- f) is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
the addF of (product G) is Relation-like K7( the carrier of (product G), the carrier of (product G)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)))
K7( the carrier of (product G), the carrier of (product G)) is Relation-like set
K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)) is Relation-like set
K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G))) is set
K462( the carrier of (product G), the addF of (product G),((G,F,f) . X),(- f)) is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
X - ((G,F) . f) is Element of the carrier of (G . F)
- ((G,F) . f) is Element of the carrier of (G . F)
X + (- ((G,F) . f)) is Element of the carrier of (G . F)
the addF of (G . F) is Relation-like K7( the carrier of (G . F), the carrier of (G . F)) -defined the carrier of (G . F) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . F), the carrier of (G . F)), the carrier of (G . F)))
K7( the carrier of (G . F), the carrier of (G . F)) is Relation-like set
K7(K7( the carrier of (G . F), the carrier of (G . F)), the carrier of (G . F)) is Relation-like set
K6(K7(K7( the carrier of (G . F), the carrier of (G . F)), the carrier of (G . F))) is set
K462( the carrier of (G . F), the addF of (G . F),X,(- ((G,F) . f))) is Element of the carrier of (G . F)
(G,F,(0. (product G))) . (X - ((G,F) . f)) is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f - ((G,F,f) . X) is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
- ((G,F,f) . X) is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f + (- ((G,F,f) . X)) is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),f,(- ((G,F,f) . X))) is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((G,F) . f) - X is Element of the carrier of (G . F)
- X is Element of the carrier of (G . F)
((G,F) . f) + (- X) is Element of the carrier of (G . F)
K462( the carrier of (G . F), the addF of (G . F),((G,F) . f),(- X)) is Element of the carrier of (G . F)
(G,F,(0. (product G))) . (((G,F) . f) - X) is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
dom (carr G) is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
r is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
dom r is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below Element of K6(NAT)
s is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
dom s is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below Element of K6(NAT)
y1 is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
dom y1 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below Element of K6(NAT)
PP is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
dom PP is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below Element of K6(NAT)
Replace (F,X,) is Relation-like Function-like set
i is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
(G,i,F) is Element of the carrier of (G . F)
G . F is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . F) is non empty set
Replace (F,(X - ((G,F) . f)),) is Relation-like Function-like set
i0 is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
DD is Element of the carrier of (G . F)
X - DD is Element of the carrier of (G . F)
- DD is Element of the carrier of (G . F)
X + (- DD) is Element of the carrier of (G . F)
K462( the carrier of (G . F), the addF of (G . F),X,(- DD)) is Element of the carrier of (G . F)
Replace (F,(X - DD),) is Relation-like Function-like set
Replace (F,(((G,F) . f) - X),) is Relation-like Function-like set
DD - X is Element of the carrier of (G . F)
DD + (- X) is Element of the carrier of (G . F)
K462( the carrier of (G . F), the addF of (G . F),DD,(- X)) is Element of the carrier of (G . F)
Replace (F,(DD - X),) is Relation-like Function-like set
F .--> X is trivial Relation-like dom G -defined {F} -defined the carrier of (G . F) -valued Function-like one-to-one constant V49() countable set
{F} is non empty trivial V49() V53() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded set
F .--> (X - DD) is trivial Relation-like dom G -defined {F} -defined the carrier of (G . F) -valued Function-like one-to-one constant V49() countable set
F .--> (DD - X) is trivial Relation-like dom G -defined {F} -defined the carrier of (G . F) -valued Function-like one-to-one constant V49() countable set
Sw1 is Relation-like Function-like set
dom Sw1 is set
w2 is set
r . w2 is set
s . w2 is set
rm is Relation-like Function-like set
dom rm is set
dom i0 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below Element of K6(NAT)
dom f is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
{F} is non empty trivial V49() V53() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(REAL)
ys is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
(G,s,ys) is Element of the carrier of (G . ys)
G . ys is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . ys) is non empty set
(G,i0,ys) is Element of the carrier of (G . ys)
y0 is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
(G,y0,ys) is Element of the carrier of (G . ys)
(G,i,ys) is Element of the carrier of (G . ys)
(G,y0,ys) - (G,i,ys) is Element of the carrier of (G . ys)
- (G,i,ys) is Element of the carrier of (G . ys)
(G,y0,ys) + (- (G,i,ys)) is Element of the carrier of (G . ys)
the addF of (G . ys) is Relation-like K7( the carrier of (G . ys), the carrier of (G . ys)) -defined the carrier of (G . ys) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . ys), the carrier of (G . ys)), the carrier of (G . ys)))
K7( the carrier of (G . ys), the carrier of (G . ys)) is Relation-like set
K7(K7( the carrier of (G . ys), the carrier of (G . ys)), the carrier of (G . ys)) is Relation-like set
K6(K7(K7( the carrier of (G . ys), the carrier of (G . ys)), the carrier of (G . ys))) is set
K462( the carrier of (G . ys), the addF of (G . ys),(G,y0,ys),(- (G,i,ys))) is Element of the carrier of (G . ys)
(G,i,ys) - (G,i,ys) is Element of the carrier of (G . ys)
(G,i,ys) + (- (G,i,ys)) is Element of the carrier of (G . ys)
K462( the carrier of (G . ys), the addF of (G . ys),(G,i,ys),(- (G,i,ys))) is Element of the carrier of (G . ys)
G . ys is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
0. (G . ys) is V68(G . ys) Element of the carrier of (G . ys)
the carrier of (G . ys) is non empty non trivial set
the ZeroF of (G . ys) is Element of the carrier of (G . ys)
{F} is non empty trivial V49() V53() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(REAL)
ys is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
(G,s,ys) is Element of the carrier of (G . ys)
G . ys is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . ys) is non empty set
y0 is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
(G,y0,ys) is Element of the carrier of (G . ys)
(G,i,ys) is Element of the carrier of (G . ys)
(G,y0,ys) - (G,i,ys) is Element of the carrier of (G . ys)
- (G,i,ys) is Element of the carrier of (G . ys)
(G,y0,ys) + (- (G,i,ys)) is Element of the carrier of (G . ys)
the addF of (G . ys) is Relation-like K7( the carrier of (G . ys), the carrier of (G . ys)) -defined the carrier of (G . ys) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . ys), the carrier of (G . ys)), the carrier of (G . ys)))
K7( the carrier of (G . ys), the carrier of (G . ys)) is Relation-like set
K7(K7( the carrier of (G . ys), the carrier of (G . ys)), the carrier of (G . ys)) is Relation-like set
K6(K7(K7( the carrier of (G . ys), the carrier of (G . ys)), the carrier of (G . ys))) is set
K462( the carrier of (G . ys), the addF of (G . ys),(G,y0,ys),(- (G,i,ys))) is Element of the carrier of (G . ys)
{F} is non empty trivial V49() V53() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(REAL)
w2 is set
y1 . w2 is set
PP . w2 is set
rm is Relation-like Function-like set
dom rm is set
dom i0 is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below Element of K6(NAT)
dom f is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
{F} is non empty trivial V49() V53() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(REAL)
ys is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
(G,PP,ys) is Element of the carrier of (G . ys)
G . ys is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . ys) is non empty set
(G,i0,ys) is Element of the carrier of (G . ys)
(G,i,ys) is Element of the carrier of (G . ys)
y0 is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
(G,y0,ys) is Element of the carrier of (G . ys)
(G,i,ys) - (G,y0,ys) is Element of the carrier of (G . ys)
- (G,y0,ys) is Element of the carrier of (G . ys)
(G,i,ys) + (- (G,y0,ys)) is Element of the carrier of (G . ys)
the addF of (G . ys) is Relation-like K7( the carrier of (G . ys), the carrier of (G . ys)) -defined the carrier of (G . ys) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . ys), the carrier of (G . ys)), the carrier of (G . ys)))
K7( the carrier of (G . ys), the carrier of (G . ys)) is Relation-like set
K7(K7( the carrier of (G . ys), the carrier of (G . ys)), the carrier of (G . ys)) is Relation-like set
K6(K7(K7( the carrier of (G . ys), the carrier of (G . ys)), the carrier of (G . ys))) is set
K462( the carrier of (G . ys), the addF of (G . ys),(G,i,ys),(- (G,y0,ys))) is Element of the carrier of (G . ys)
(G,i,ys) - (G,i,ys) is Element of the carrier of (G . ys)
- (G,i,ys) is Element of the carrier of (G . ys)
(G,i,ys) + (- (G,i,ys)) is Element of the carrier of (G . ys)
K462( the carrier of (G . ys), the addF of (G . ys),(G,i,ys),(- (G,i,ys))) is Element of the carrier of (G . ys)
G . ys is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
0. (G . ys) is V68(G . ys) Element of the carrier of (G . ys)
the carrier of (G . ys) is non empty non trivial set
the ZeroF of (G . ys) is Element of the carrier of (G . ys)
{F} is non empty trivial V49() V53() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(REAL)
ys is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
(G,PP,ys) is Element of the carrier of (G . ys)
G . ys is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . ys) is non empty set
(G,i,ys) is Element of the carrier of (G . ys)
y0 is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
(G,y0,ys) is Element of the carrier of (G . ys)
(G,i,ys) - (G,y0,ys) is Element of the carrier of (G . ys)
- (G,y0,ys) is Element of the carrier of (G . ys)
(G,i,ys) + (- (G,y0,ys)) is Element of the carrier of (G . ys)
the addF of (G . ys) is Relation-like K7( the carrier of (G . ys), the carrier of (G . ys)) -defined the carrier of (G . ys) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . ys), the carrier of (G . ys)), the carrier of (G . ys)))
K7( the carrier of (G . ys), the carrier of (G . ys)) is Relation-like set
K7(K7( the carrier of (G . ys), the carrier of (G . ys)), the carrier of (G . ys)) is Relation-like set
K6(K7(K7( the carrier of (G . ys), the carrier of (G . ys)), the carrier of (G . ys))) is set
K462( the carrier of (G . ys), the addF of (G . ys),(G,i,ys),(- (G,y0,ys))) is Element of the carrier of (G . ys)
{F} is non empty trivial V49() V53() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(REAL)
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
K6( the carrier of (product G)) is set
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . F) is non empty non trivial set
(G,F) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . F) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . F)))
K7( the carrier of (product G), the carrier of (G . F)) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . F))) is set
f is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,F) . f is Element of the carrier of (G . F)
(G,F,f) is non empty Relation-like the carrier of (G . F) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . F), the carrier of (product G)))
K7( the carrier of (G . F), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . F), the carrier of (product G))) is set
X is Element of K6( the carrier of (product G))
m is V11() real ext-real Element of REAL
{ b₁ where b₁ is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G) : not m <= ||.(b₁ - f).|| } is set
{ b₁ where b₁ is Element of the carrier of (G . F) : not m <= ||.(b₁ - ((G,F) . f)).|| } is set
i0 is Neighbourhood of (G,F) . f
y0 is Element of the carrier of (G . F)
(G,F,f) . y0 is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((G,F,f) . y0) - f is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
- f is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((G,F,f) . y0) + (- f) is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
the addF of (product G) is Relation-like K7( the carrier of (product G), the carrier of (product G)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)))
K7( the carrier of (product G), the carrier of (product G)) is Relation-like set
K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)) is Relation-like set
K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G))) is set
K462( the carrier of (product G), the addF of (product G),((G,F,f) . y0),(- f)) is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.(((G,F,f) . y0) - f).|| is V11() real ext-real non negative Element of REAL
the normF of (product G) is non empty Relation-like the carrier of (product G) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (product G),REAL))
K7( the carrier of (product G),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (product G),REAL)) is non trivial V49() set
the normF of (product G) . (((G,F,f) . y0) - f) is V11() real ext-real Element of REAL
0. (product G) is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like V68( product G) countable Element of the carrier of (product G)
the ZeroF of (product G) is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,F,(0. (product G))) is non empty Relation-like the carrier of (G . F) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . F), the carrier of (product G)))
y0 - ((G,F) . f) is Element of the carrier of (G . F)
- ((G,F) . f) is Element of the carrier of (G . F)
y0 + (- ((G,F) . f)) is Element of the carrier of (G . F)
the addF of (G . F) is Relation-like K7( the carrier of (G . F), the carrier of (G . F)) -defined the carrier of (G . F) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . F), the carrier of (G . F)), the carrier of (G . F)))
K7( the carrier of (G . F), the carrier of (G . F)) is Relation-like set
K7(K7( the carrier of (G . F), the carrier of (G . F)), the carrier of (G . F)) is Relation-like set
K6(K7(K7( the carrier of (G . F), the carrier of (G . F)), the carrier of (G . F))) is set
K462( the carrier of (G . F), the addF of (G . F),y0,(- ((G,F) . f))) is Element of the carrier of (G . F)
(G,F,(0. (product G))) . (y0 - ((G,F) . f)) is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.((G,F,(0. (product G))) . (y0 - ((G,F) . f))).|| is V11() real ext-real non negative Element of REAL
the normF of (product G) . ((G,F,(0. (product G))) . (y0 - ((G,F) . f))) is V11() real ext-real Element of REAL
||.(y0 - ((G,F) . f)).|| is V11() real ext-real non negative Element of REAL
the normF of (G . F) is non empty Relation-like the carrier of (G . F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (G . F),REAL))
K7( the carrier of (G . F),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (G . F),REAL)) is non trivial V49() set
the normF of (G . F) . (y0 - ((G,F) . f)) is V11() real ext-real Element of REAL
r is Element of the carrier of (G . F)
r - ((G,F) . f) is Element of the carrier of (G . F)
r + (- ((G,F) . f)) is Element of the carrier of (G . F)
K462( the carrier of (G . F), the addF of (G . F),r,(- ((G,F) . f))) is Element of the carrier of (G . F)
||.(r - ((G,F) . f)).|| is V11() real ext-real non negative Element of REAL
the normF of (G . F) . (r - ((G,F) . f)) is V11() real ext-real Element of REAL
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
K6( the carrier of (product G)) is set
F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of F is non empty non trivial set
K7( the carrier of (product G), the carrier of F) is Relation-like set
K6(K7( the carrier of (product G), the carrier of F)) is set
f is set
X is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of F))
dom X is Element of K6( the carrier of (product G))
m is Element of K6( the carrier of (product G))
(G,f) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
G . (G,f) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,f)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,f)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,f)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,f)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,f)),F))
the carrier of (G . (G,f)) is non empty non trivial set
K266( the carrier of (G . (G,f)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,f)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,f)), the carrier of F
FuncZero ( the carrier of (G . (G,f)),F) is Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,f)), the carrier of F)
0. F is V68(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
the carrier of (G . (G,f)) --> (0. F) is non empty Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,f)), the carrier of F))
K7( the carrier of (G . (G,f)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,f)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,f)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,f)), the carrier of F)),(Funcs ( the carrier of (G . (G,f)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,f)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,f)), the carrier of F)),(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,f)), the carrier of F)),(Funcs ( the carrier of (G . (G,f)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,f)), the carrier of F)),(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,f)), the carrier of F)),(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,f)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,f)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,f)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,f)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,f)), the carrier of F)),(FuncZero ( the carrier of (G . (G,f)),F)),(FuncAdd ( the carrier of (G . (G,f)),F)),(FuncExtMult ( the carrier of (G . (G,f)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,f)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,f)),F)) is set
K215(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,f)),F)
K213(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))) is Relation-like K7((LinearOperators ((G . (G,f)),F)),(LinearOperators ((G . (G,f)),F))) -defined LinearOperators ((G . (G,f)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,f)),F)),(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F))))
K7((LinearOperators ((G . (G,f)),F)),(LinearOperators ((G . (G,f)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,f)),F)),(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,f)),F)),(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F)))) is set
K214(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,f)),F))) -defined LinearOperators ((G . (G,f)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F))))
K7(REAL,(LinearOperators ((G . (G,f)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,f)),F)),K215(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))),K213(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))),K214(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,f)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,f)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))) is Element of BoundedLinearOperators ((G . (G,f)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))) -defined BoundedLinearOperators ((G . (G,f)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F))))
K7((BoundedLinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,f)),F))) -defined BoundedLinearOperators ((G . (G,f)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,f)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,f)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,f)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,f)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,f)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,f)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,f)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperatorsNorm ((G . (G,f)),F)) #) is strict NORMSTR
r is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,(G,f)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,f)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,f))))
K7( the carrier of (product G), the carrier of (G . (G,f))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,f)))) is set
(G,(G,f)) . r is Element of the carrier of (G . (G,f))
X | m is Relation-like the carrier of (product G) -defined m -defined the carrier of (product G) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of F))
(G,(G,f),r) is non empty Relation-like the carrier of (G . (G,f)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,f)), the carrier of (product G)))
K7( the carrier of (G . (G,f)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,f)), the carrier of (product G))) is set
(X | m) * (G,(G,f),r) is Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,f)), the carrier of F))
s is Element of the carrier of (G . (G,f))
dom ((X | m) * (G,(G,f),r)) is Element of K6( the carrier of (G . (G,f)))
K6( the carrier of (G . (G,f))) is set
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F)) is non empty set
((X | m) * (G,(G,f),r)) /. s is Element of the carrier of F
y1 is Neighbourhood of s
PP is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F))
DD is Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like RestFunc-like Element of K6(K7( the carrier of (G . (G,f)), the carrier of F))
PP is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F))
DD is Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like RestFunc-like Element of K6(K7( the carrier of (G . (G,f)), the carrier of F))
a is Neighbourhood of s
t is Element of the carrier of (G . (G,f))
(G,(G,f),r) . t is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(dom X) /\ m is Element of K6( the carrier of (product G))
dom (X | m) is Element of K6(m)
K6(m) is set
y1 /\ a is Element of K6( the carrier of (G . (G,f)))
X * (G,(G,f),r) is Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,f)), the carrier of F))
dom (X * (G,(G,f),r)) is Element of K6( the carrier of (G . (G,f)))
t is Neighbourhood of s
tm is Element of the carrier of (G . (G,f))
dom (G,(G,f),r) is non empty Element of K6( the carrier of (G . (G,f)))
(G,(G,f),r) . tm is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,(G,f),r) . s is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((X | m) * (G,(G,f),r)) /. tm is Element of the carrier of F
(G,(G,f),r) /. tm is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(X | m) /. ((G,(G,f),r) /. tm) is Element of the carrier of F
X /. ((G,(G,f),r) /. tm) is Element of the carrier of F
(X * (G,(G,f),r)) /. tm is Element of the carrier of F
(G,(G,f),r) /. s is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(X | m) /. ((G,(G,f),r) /. s) is Element of the carrier of F
X /. ((G,(G,f),r) /. s) is Element of the carrier of F
(X * (G,(G,f),r)) /. s is Element of the carrier of F
((X * (G,(G,f),r)) /. tm) - ((X * (G,(G,f),r)) /. s) is Element of the carrier of F
- ((X * (G,(G,f),r)) /. s) is Element of the carrier of F
((X * (G,(G,f),r)) /. tm) + (- ((X * (G,(G,f),r)) /. s)) is Element of the carrier of F
the addF of F is Relation-like K7( the carrier of F, the carrier of F) -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of F, the carrier of F), the carrier of F))
K7( the carrier of F, the carrier of F) is Relation-like set
K7(K7( the carrier of F, the carrier of F), the carrier of F) is Relation-like set
K6(K7(K7( the carrier of F, the carrier of F), the carrier of F)) is set
K462( the carrier of F, the addF of F,((X * (G,(G,f),r)) /. tm),(- ((X * (G,(G,f),r)) /. s))) is Element of the carrier of F
tm - s is Element of the carrier of (G . (G,f))
- s is Element of the carrier of (G . (G,f))
tm + (- s) is Element of the carrier of (G . (G,f))
the addF of (G . (G,f)) is Relation-like K7( the carrier of (G . (G,f)), the carrier of (G . (G,f))) -defined the carrier of (G . (G,f)) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . (G,f)), the carrier of (G . (G,f))), the carrier of (G . (G,f))))
K7( the carrier of (G . (G,f)), the carrier of (G . (G,f))) is Relation-like set
K7(K7( the carrier of (G . (G,f)), the carrier of (G . (G,f))), the carrier of (G . (G,f))) is Relation-like set
K6(K7(K7( the carrier of (G . (G,f)), the carrier of (G . (G,f))), the carrier of (G . (G,f)))) is set
K462( the carrier of (G . (G,f)), the addF of (G . (G,f)),tm,(- s)) is Element of the carrier of (G . (G,f))
PP . (tm - s) is Element of the carrier of F
DD /. (tm - s) is Element of the carrier of F
(PP . (tm - s)) + (DD /. (tm - s)) is Element of the carrier of F
K462( the carrier of F, the addF of F,(PP . (tm - s)),(DD /. (tm - s))) is Element of the carrier of F
r is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,(G,f)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,f)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,f))))
K7( the carrier of (product G), the carrier of (G . (G,f))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,f)))) is set
(G,(G,f)) . r is Element of the carrier of (G . (G,f))
(G,(G,f),r) is non empty Relation-like the carrier of (G . (G,f)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,f)), the carrier of (product G)))
K7( the carrier of (G . (G,f)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,f)), the carrier of (product G))) is set
X * (G,(G,f),r) is Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,f)), the carrier of F))
s is Element of the carrier of (G . (G,f))
dom (X * (G,(G,f),r)) is Element of K6( the carrier of (G . (G,f)))
K6( the carrier of (G . (G,f))) is set
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F)) is non empty set
(X * (G,(G,f),r)) /. s is Element of the carrier of F
y1 is Neighbourhood of s
PP is Neighbourhood of s
DD is Element of the carrier of (G . (G,f))
(G,(G,f),r) . DD is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(dom X) /\ m is Element of K6( the carrier of (product G))
X | m is Relation-like the carrier of (product G) -defined m -defined the carrier of (product G) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of F))
dom (X | m) is Element of K6(m)
K6(m) is set
(X | m) * (G,(G,f),r) is Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,f)), the carrier of F))
dom ((X | m) * (G,(G,f),r)) is Element of K6( the carrier of (G . (G,f)))
DD is set
a is Element of the carrier of (G . (G,f))
(G,(G,f),r) . a is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
dom (G,(G,f),r) is non empty Element of K6( the carrier of (G . (G,f)))
y1 /\ PP is Element of K6( the carrier of (G . (G,f)))
DD is Neighbourhood of s
a is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F))
t is Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like RestFunc-like Element of K6(K7( the carrier of (G . (G,f)), the carrier of F))
a is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F))
t is Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like RestFunc-like Element of K6(K7( the carrier of (G . (G,f)), the carrier of F))
tm is Element of the carrier of (G . (G,f))
dom (G,(G,f),r) is non empty Element of K6( the carrier of (G . (G,f)))
(G,(G,f),r) . tm is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,(G,f),r) . s is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((X | m) * (G,(G,f),r)) /. tm is Element of the carrier of F
(G,(G,f),r) /. tm is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(X | m) /. ((G,(G,f),r) /. tm) is Element of the carrier of F
X /. ((G,(G,f),r) /. tm) is Element of the carrier of F
(X * (G,(G,f),r)) /. tm is Element of the carrier of F
((X | m) * (G,(G,f),r)) /. s is Element of the carrier of F
(G,(G,f),r) /. s is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(X | m) /. ((G,(G,f),r) /. s) is Element of the carrier of F
X /. ((G,(G,f),r) /. s) is Element of the carrier of F
(((X | m) * (G,(G,f),r)) /. tm) - (((X | m) * (G,(G,f),r)) /. s) is Element of the carrier of F
- (((X | m) * (G,(G,f),r)) /. s) is Element of the carrier of F
(((X | m) * (G,(G,f),r)) /. tm) + (- (((X | m) * (G,(G,f),r)) /. s)) is Element of the carrier of F
the addF of F is Relation-like K7( the carrier of F, the carrier of F) -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of F, the carrier of F), the carrier of F))
K7( the carrier of F, the carrier of F) is Relation-like set
K7(K7( the carrier of F, the carrier of F), the carrier of F) is Relation-like set
K6(K7(K7( the carrier of F, the carrier of F), the carrier of F)) is set
K462( the carrier of F, the addF of F,(((X | m) * (G,(G,f),r)) /. tm),(- (((X | m) * (G,(G,f),r)) /. s))) is Element of the carrier of F
tm - s is Element of the carrier of (G . (G,f))
- s is Element of the carrier of (G . (G,f))
tm + (- s) is Element of the carrier of (G . (G,f))
the addF of (G . (G,f)) is Relation-like K7( the carrier of (G . (G,f)), the carrier of (G . (G,f))) -defined the carrier of (G . (G,f)) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . (G,f)), the carrier of (G . (G,f))), the carrier of (G . (G,f))))
K7( the carrier of (G . (G,f)), the carrier of (G . (G,f))) is Relation-like set
K7(K7( the carrier of (G . (G,f)), the carrier of (G . (G,f))), the carrier of (G . (G,f))) is Relation-like set
K6(K7(K7( the carrier of (G . (G,f)), the carrier of (G . (G,f))), the carrier of (G . (G,f)))) is set
K462( the carrier of (G . (G,f)), the addF of (G . (G,f)),tm,(- s)) is Element of the carrier of (G . (G,f))
a . (tm - s) is Element of the carrier of F
t /. (tm - s) is Element of the carrier of F
(a . (tm - s)) + (t /. (tm - s)) is Element of the carrier of F
K462( the carrier of F, the addF of F,(a . (tm - s)),(t /. (tm - s))) is Element of the carrier of F
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
K6( the carrier of (product G)) is set
F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of F is non empty non trivial set
K7( the carrier of (product G), the carrier of F) is Relation-like set
K6(K7( the carrier of (product G), the carrier of F)) is set
f is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of F))
X is set
m is set
dom f is Element of K6( the carrier of (product G))
f is set
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of F is non empty non trivial set
K7( the carrier of (product G), the carrier of F) is Relation-like set
K6(K7( the carrier of (product G), the carrier of F)) is set
X is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of F))
m is set
(G,f) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . (G,f) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,f)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,f)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,f)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,f)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,f)),F))
the carrier of (G . (G,f)) is non empty non trivial set
K266( the carrier of (G . (G,f)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,f)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,f)), the carrier of F
FuncZero ( the carrier of (G . (G,f)),F) is Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,f)), the carrier of F)
0. F is V68(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
the carrier of (G . (G,f)) --> (0. F) is non empty Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,f)), the carrier of F))
K7( the carrier of (G . (G,f)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,f)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,f)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,f)), the carrier of F)),(Funcs ( the carrier of (G . (G,f)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,f)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,f)), the carrier of F)),(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,f)), the carrier of F)),(Funcs ( the carrier of (G . (G,f)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,f)), the carrier of F)),(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,f)), the carrier of F)),(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,f)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,f)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,f)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,f)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,f)), the carrier of F)),(FuncZero ( the carrier of (G . (G,f)),F)),(FuncAdd ( the carrier of (G . (G,f)),F)),(FuncExtMult ( the carrier of (G . (G,f)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,f)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,f)),F)) is set
K215(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,f)),F)
K213(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))) is Relation-like K7((LinearOperators ((G . (G,f)),F)),(LinearOperators ((G . (G,f)),F))) -defined LinearOperators ((G . (G,f)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,f)),F)),(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F))))
K7((LinearOperators ((G . (G,f)),F)),(LinearOperators ((G . (G,f)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,f)),F)),(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,f)),F)),(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F)))) is set
K214(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,f)),F))) -defined LinearOperators ((G . (G,f)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F))))
K7(REAL,(LinearOperators ((G . (G,f)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,f)),F)),K215(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))),K213(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))),K214(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,f)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,f)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))) is Element of BoundedLinearOperators ((G . (G,f)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))) -defined BoundedLinearOperators ((G . (G,f)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F))))
K7((BoundedLinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,f)),F))) -defined BoundedLinearOperators ((G . (G,f)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,f)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,f)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,f)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,f)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,f)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,f)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,f)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperatorsNorm ((G . (G,f)),F)) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F)) is non empty set
K7( the carrier of (product G), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F)))) is set
i is Relation-like the carrier of (product G) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F)) -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F))))
dom i is Element of K6( the carrier of (product G))
K6( the carrier of (product G)) is set
i0 is set
i0 is set
i0 is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
i . i0 is set
(G,F,f,X,i0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F))
(G,(G,f),i0) is non empty Relation-like the carrier of (G . (G,f)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,f)), the carrier of (product G)))
K7( the carrier of (G . (G,f)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,f)), the carrier of (product G))) is set
X * (G,(G,f),i0) is Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,f)), the carrier of F))
(G,(G,f)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,f)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,f))))
K7( the carrier of (product G), the carrier of (G . (G,f))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,f)))) is set
(G,(G,f)) . i0 is Element of the carrier of (G . (G,f))
diff ((X * (G,(G,f),i0)),((G,(G,f)) . i0)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F))
i /. i0 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F))
i is Relation-like the carrier of (product G) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F)) -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F))))
dom i is Element of K6( the carrier of (product G))
K6( the carrier of (product G)) is set
i0 is Relation-like the carrier of (product G) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F)) -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F))))
dom i0 is Element of K6( the carrier of (product G))
y0 is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
i /. y0 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F))
(G,F,f,X,y0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F))
(G,(G,f),y0) is non empty Relation-like the carrier of (G . (G,f)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,f)), the carrier of (product G)))
K7( the carrier of (G . (G,f)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,f)), the carrier of (product G))) is set
X * (G,(G,f),y0) is Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,f)), the carrier of F))
(G,(G,f)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,f)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,f))))
K7( the carrier of (product G), the carrier of (G . (G,f))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,f)))) is set
(G,(G,f)) . y0 is Element of the carrier of (G . (G,f))
diff ((X * (G,(G,f),y0)),((G,(G,f)) . y0)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F))
i0 /. y0 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F))
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of F is non empty non trivial set
K7( the carrier of (product G), the carrier of F) is Relation-like set
K6(K7( the carrier of (product G), the carrier of F)) is set
f is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of F))
X is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of F))
f + X is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of F))
f - X is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of F))
m is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
i is set
(G,i) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . (G,i) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . (G,i)) is non empty non trivial set
(G,(G,i),m) is non empty Relation-like the carrier of (G . (G,i)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,i)), the carrier of (product G)))
K7( the carrier of (G . (G,i)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,i)), the carrier of (product G))) is set
(f + X) * (G,(G,i),m) is Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,i)), the carrier of F))
K7( the carrier of (G . (G,i)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,i)), the carrier of F)) is set
f * (G,(G,i),m) is Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,i)), the carrier of F))
X * (G,(G,i),m) is Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,i)), the carrier of F))
(f * (G,(G,i),m)) + (X * (G,(G,i),m)) is Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,i)), the carrier of F))
(f - X) * (G,(G,i),m) is Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,i)), the carrier of F))
(f * (G,(G,i),m)) - (X * (G,(G,i),m)) is Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,i)), the carrier of F))
dom (G,(G,i),m) is non empty Element of K6( the carrier of (G . (G,i)))
K6( the carrier of (G . (G,i))) is set
dom (f + X) is Element of K6( the carrier of (product G))
K6( the carrier of (product G)) is set
dom f is Element of K6( the carrier of (product G))
dom X is Element of K6( the carrier of (product G))
(dom f) /\ (dom X) is Element of K6( the carrier of (product G))
dom ((f + X) * (G,(G,i),m)) is Element of K6( the carrier of (G . (G,i)))
dom ((f * (G,(G,i),m)) + (X * (G,(G,i),m))) is Element of K6( the carrier of (G . (G,i)))
y0 is Element of the carrier of (G . (G,i))
(G,(G,i),m) . y0 is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
dom (f * (G,(G,i),m)) is Element of K6( the carrier of (G . (G,i)))
dom (X * (G,(G,i),m)) is Element of K6( the carrier of (G . (G,i)))
(dom (f * (G,(G,i),m))) /\ (dom (X * (G,(G,i),m))) is Element of K6( the carrier of (G . (G,i)))
y0 is set
r is set
y0 is Element of the carrier of (G . (G,i))
((f + X) * (G,(G,i),m)) . y0 is set
((f * (G,(G,i),m)) + (X * (G,(G,i),m))) . y0 is set
(G,(G,i),m) . y0 is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
dom (f * (G,(G,i),m)) is Element of K6( the carrier of (G . (G,i)))
dom (X * (G,(G,i),m)) is Element of K6( the carrier of (G . (G,i)))
(dom (f * (G,(G,i),m))) /\ (dom (X * (G,(G,i),m))) is Element of K6( the carrier of (G . (G,i)))
f /. ((G,(G,i),m) . y0) is Element of the carrier of F
f . ((G,(G,i),m) . y0) is set
(f * (G,(G,i),m)) . y0 is set
(f * (G,(G,i),m)) /. y0 is Element of the carrier of F
X /. ((G,(G,i),m) . y0) is Element of the carrier of F
X . ((G,(G,i),m) . y0) is set
(X * (G,(G,i),m)) . y0 is set
(X * (G,(G,i),m)) /. y0 is Element of the carrier of F
(f + X) . ((G,(G,i),m) . y0) is set
(f + X) /. ((G,(G,i),m) . y0) is Element of the carrier of F
(f /. ((G,(G,i),m) . y0)) + (X /. ((G,(G,i),m) . y0)) is Element of the carrier of F
the addF of F is Relation-like K7( the carrier of F, the carrier of F) -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of F, the carrier of F), the carrier of F))
K7( the carrier of F, the carrier of F) is Relation-like set
K7(K7( the carrier of F, the carrier of F), the carrier of F) is Relation-like set
K6(K7(K7( the carrier of F, the carrier of F), the carrier of F)) is set
K462( the carrier of F, the addF of F,(f /. ((G,(G,i),m) . y0)),(X /. ((G,(G,i),m) . y0))) is Element of the carrier of F
((f * (G,(G,i),m)) + (X * (G,(G,i),m))) /. y0 is Element of the carrier of F
dom (f - X) is Element of K6( the carrier of (product G))
dom ((f - X) * (G,(G,i),m)) is Element of K6( the carrier of (G . (G,i)))
dom ((f * (G,(G,i),m)) - (X * (G,(G,i),m))) is Element of K6( the carrier of (G . (G,i)))
y0 is Element of the carrier of (G . (G,i))
(G,(G,i),m) . y0 is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
dom (f * (G,(G,i),m)) is Element of K6( the carrier of (G . (G,i)))
dom (X * (G,(G,i),m)) is Element of K6( the carrier of (G . (G,i)))
(dom (f * (G,(G,i),m))) /\ (dom (X * (G,(G,i),m))) is Element of K6( the carrier of (G . (G,i)))
y0 is set
r is set
y0 is Element of the carrier of (G . (G,i))
((f - X) * (G,(G,i),m)) . y0 is set
((f * (G,(G,i),m)) - (X * (G,(G,i),m))) . y0 is set
(G,(G,i),m) . y0 is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
dom (f * (G,(G,i),m)) is Element of K6( the carrier of (G . (G,i)))
dom (X * (G,(G,i),m)) is Element of K6( the carrier of (G . (G,i)))
(dom (f * (G,(G,i),m))) /\ (dom (X * (G,(G,i),m))) is Element of K6( the carrier of (G . (G,i)))
f /. ((G,(G,i),m) . y0) is Element of the carrier of F
f . ((G,(G,i),m) . y0) is set
(f * (G,(G,i),m)) . y0 is set
(f * (G,(G,i),m)) /. y0 is Element of the carrier of F
X /. ((G,(G,i),m) . y0) is Element of the carrier of F
X . ((G,(G,i),m) . y0) is set
(X * (G,(G,i),m)) . y0 is set
(X * (G,(G,i),m)) /. y0 is Element of the carrier of F
(f - X) . ((G,(G,i),m) . y0) is set
(f - X) /. ((G,(G,i),m) . y0) is Element of the carrier of F
(f /. ((G,(G,i),m) . y0)) - (X /. ((G,(G,i),m) . y0)) is Element of the carrier of F
- (X /. ((G,(G,i),m) . y0)) is Element of the carrier of F
(f /. ((G,(G,i),m) . y0)) + (- (X /. ((G,(G,i),m) . y0))) is Element of the carrier of F
the addF of F is Relation-like K7( the carrier of F, the carrier of F) -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of F, the carrier of F), the carrier of F))
K7( the carrier of F, the carrier of F) is Relation-like set
K7(K7( the carrier of F, the carrier of F), the carrier of F) is Relation-like set
K6(K7(K7( the carrier of F, the carrier of F), the carrier of F)) is set
K462( the carrier of F, the addF of F,(f /. ((G,(G,i),m) . y0)),(- (X /. ((G,(G,i),m) . y0)))) is Element of the carrier of F
((f * (G,(G,i),m)) - (X * (G,(G,i),m))) /. y0 is Element of the carrier of F
G is V11() real ext-real Element of REAL
F is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
product F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product F) is non empty non trivial set
dom F is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
f is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of f is non empty non trivial set
K7( the carrier of (product F), the carrier of f) is Relation-like set
K6(K7( the carrier of (product F), the carrier of f)) is set
X is Relation-like the carrier of (product F) -defined the carrier of f -valued Function-like Element of K6(K7( the carrier of (product F), the carrier of f))
G (#) X is Relation-like the carrier of (product F) -defined the carrier of f -valued Function-like Element of K6(K7( the carrier of (product F), the carrier of f))
m is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product F)
i is set
(F,i) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom F
F . (F,i) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (F . (F,i)) is non empty non trivial set
(F,(F,i),m) is non empty Relation-like the carrier of (F . (F,i)) -defined the carrier of (product F) -valued Function-like total quasi_total Element of K6(K7( the carrier of (F . (F,i)), the carrier of (product F)))
K7( the carrier of (F . (F,i)), the carrier of (product F)) is Relation-like set
K6(K7( the carrier of (F . (F,i)), the carrier of (product F))) is set
X * (F,(F,i),m) is Relation-like the carrier of (F . (F,i)) -defined the carrier of f -valued Function-like Element of K6(K7( the carrier of (F . (F,i)), the carrier of f))
K7( the carrier of (F . (F,i)), the carrier of f) is Relation-like set
K6(K7( the carrier of (F . (F,i)), the carrier of f)) is set
G (#) (X * (F,(F,i),m)) is Relation-like the carrier of (F . (F,i)) -defined the carrier of f -valued Function-like Element of K6(K7( the carrier of (F . (F,i)), the carrier of f))
(G (#) X) * (F,(F,i),m) is Relation-like the carrier of (F . (F,i)) -defined the carrier of f -valued Function-like Element of K6(K7( the carrier of (F . (F,i)), the carrier of f))
dom (G (#) X) is Element of K6( the carrier of (product F))
K6( the carrier of (product F)) is set
dom X is Element of K6( the carrier of (product F))
dom (G (#) (X * (F,(F,i),m))) is Element of K6( the carrier of (F . (F,i)))
K6( the carrier of (F . (F,i))) is set
dom (X * (F,(F,i),m)) is Element of K6( the carrier of (F . (F,i)))
dom (F,(F,i),m) is non empty Element of K6( the carrier of (F . (F,i)))
dom ((G (#) X) * (F,(F,i),m)) is Element of K6( the carrier of (F . (F,i)))
y0 is Element of the carrier of (F . (F,i))
(F,(F,i),m) . y0 is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product F)
y0 is set
r is set
y0 is Element of the carrier of (F . (F,i))
(F,(F,i),m) . y0 is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product F)
y0 is Element of the carrier of (F . (F,i))
(G (#) (X * (F,(F,i),m))) . y0 is set
((G (#) X) * (F,(F,i),m)) . y0 is set
(F,(F,i),m) . y0 is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product F)
X /. ((F,(F,i),m) . y0) is Element of the carrier of f
X . ((F,(F,i),m) . y0) is set
(X * (F,(F,i),m)) . y0 is set
(X * (F,(F,i),m)) /. y0 is Element of the carrier of f
(G (#) (X * (F,(F,i),m))) /. y0 is Element of the carrier of f
G * (X /. ((F,(F,i),m) . y0)) is Element of the carrier of f
the Mult of f is Relation-like K7(REAL, the carrier of f) -defined the carrier of f -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of f), the carrier of f))
K7(REAL, the carrier of f) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of f), the carrier of f) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of f), the carrier of f)) is non trivial V49() set
K458( the Mult of f,G,(X /. ((F,(F,i),m) . y0))) is set
(G (#) X) . ((F,(F,i),m) . y0) is set
(G (#) X) /. ((F,(F,i),m) . y0) is Element of the carrier of f
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of F is non empty non trivial set
K7( the carrier of (product G), the carrier of F) is Relation-like set
K6(K7( the carrier of (product G), the carrier of F)) is set
f is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of F))
X is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of F))
f + X is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of F))
m is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
i is set
(G,F,i,(f + X),m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
(G,i) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . (G,i) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,i)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,i)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,i)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,i)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,i)),F))
the carrier of (G . (G,i)) is non empty non trivial set
K266( the carrier of (G . (G,i)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,i)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,i)), the carrier of F
FuncZero ( the carrier of (G . (G,i)),F) is Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,i)), the carrier of F)
0. F is V68(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
the carrier of (G . (G,i)) --> (0. F) is non empty Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,i)), the carrier of F))
K7( the carrier of (G . (G,i)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,i)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,i)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,i)), the carrier of F)),(Funcs ( the carrier of (G . (G,i)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,i)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,i)), the carrier of F)),(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,i)), the carrier of F)),(Funcs ( the carrier of (G . (G,i)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,i)), the carrier of F)),(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,i)), the carrier of F)),(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,i)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,i)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,i)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,i)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,i)), the carrier of F)),(FuncZero ( the carrier of (G . (G,i)),F)),(FuncAdd ( the carrier of (G . (G,i)),F)),(FuncExtMult ( the carrier of (G . (G,i)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,i)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,i)),F)) is set
K215(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,i)),F)
K213(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))) is Relation-like K7((LinearOperators ((G . (G,i)),F)),(LinearOperators ((G . (G,i)),F))) -defined LinearOperators ((G . (G,i)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,i)),F)),(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F))))
K7((LinearOperators ((G . (G,i)),F)),(LinearOperators ((G . (G,i)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,i)),F)),(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,i)),F)),(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F)))) is set
K214(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,i)),F))) -defined LinearOperators ((G . (G,i)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F))))
K7(REAL,(LinearOperators ((G . (G,i)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,i)),F)),K215(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))),K213(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))),K214(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,i)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,i)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))) is Element of BoundedLinearOperators ((G . (G,i)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))) -defined BoundedLinearOperators ((G . (G,i)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F))))
K7((BoundedLinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,i)),F))) -defined BoundedLinearOperators ((G . (G,i)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,i)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,i)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,i)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,i)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,i)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,i)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,i)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperatorsNorm ((G . (G,i)),F)) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)) is non empty set
(G,(G,i),m) is non empty Relation-like the carrier of (G . (G,i)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,i)), the carrier of (product G)))
K7( the carrier of (G . (G,i)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,i)), the carrier of (product G))) is set
(f + X) * (G,(G,i),m) is Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,i)), the carrier of F))
(G,(G,i)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,i)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,i))))
K7( the carrier of (product G), the carrier of (G . (G,i))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,i)))) is set
(G,(G,i)) . m is Element of the carrier of (G . (G,i))
diff (((f + X) * (G,(G,i),m)),((G,(G,i)) . m)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
(G,F,i,f,m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
f * (G,(G,i),m) is Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,i)), the carrier of F))
diff ((f * (G,(G,i),m)),((G,(G,i)) . m)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
(G,F,i,X,m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
X * (G,(G,i),m) is Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,i)), the carrier of F))
diff ((X * (G,(G,i),m)),((G,(G,i)) . m)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
(G,F,i,f,m) + (G,F,i,X,m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
the addF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)) is Relation-like K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))) is Relation-like set
K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))) is Relation-like set
K6(K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)))) is set
K462( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)), the addF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)),(G,F,i,f,m),(G,F,i,X,m)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
(f * (G,(G,i),m)) + (X * (G,(G,i),m)) is Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,i)), the carrier of F))
diff (((f * (G,(G,i),m)) + (X * (G,(G,i),m))),((G,(G,i)) . m)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of F is non empty non trivial set
K7( the carrier of (product G), the carrier of F) is Relation-like set
K6(K7( the carrier of (product G), the carrier of F)) is set
f is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of F))
X is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of F))
f - X is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of F))
m is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
i is set
(G,F,i,(f - X),m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
(G,i) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . (G,i) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,i)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,i)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,i)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,i)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,i)),F))
the carrier of (G . (G,i)) is non empty non trivial set
K266( the carrier of (G . (G,i)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,i)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,i)), the carrier of F
FuncZero ( the carrier of (G . (G,i)),F) is Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,i)), the carrier of F)
0. F is V68(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
the carrier of (G . (G,i)) --> (0. F) is non empty Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,i)), the carrier of F))
K7( the carrier of (G . (G,i)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,i)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,i)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,i)), the carrier of F)),(Funcs ( the carrier of (G . (G,i)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,i)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,i)), the carrier of F)),(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,i)), the carrier of F)),(Funcs ( the carrier of (G . (G,i)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,i)), the carrier of F)),(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,i)), the carrier of F)),(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,i)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,i)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,i)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,i)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,i)), the carrier of F)),(FuncZero ( the carrier of (G . (G,i)),F)),(FuncAdd ( the carrier of (G . (G,i)),F)),(FuncExtMult ( the carrier of (G . (G,i)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,i)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,i)),F)) is set
K215(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,i)),F)
K213(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))) is Relation-like K7((LinearOperators ((G . (G,i)),F)),(LinearOperators ((G . (G,i)),F))) -defined LinearOperators ((G . (G,i)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,i)),F)),(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F))))
K7((LinearOperators ((G . (G,i)),F)),(LinearOperators ((G . (G,i)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,i)),F)),(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,i)),F)),(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F)))) is set
K214(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,i)),F))) -defined LinearOperators ((G . (G,i)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F))))
K7(REAL,(LinearOperators ((G . (G,i)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,i)),F)),K215(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))),K213(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))),K214(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,i)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,i)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))) is Element of BoundedLinearOperators ((G . (G,i)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))) -defined BoundedLinearOperators ((G . (G,i)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F))))
K7((BoundedLinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,i)),F))) -defined BoundedLinearOperators ((G . (G,i)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,i)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,i)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,i)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,i)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,i)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,i)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,i)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperatorsNorm ((G . (G,i)),F)) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)) is non empty set
(G,(G,i),m) is non empty Relation-like the carrier of (G . (G,i)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,i)), the carrier of (product G)))
K7( the carrier of (G . (G,i)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,i)), the carrier of (product G))) is set
(f - X) * (G,(G,i),m) is Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,i)), the carrier of F))
(G,(G,i)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,i)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,i))))
K7( the carrier of (product G), the carrier of (G . (G,i))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,i)))) is set
(G,(G,i)) . m is Element of the carrier of (G . (G,i))
diff (((f - X) * (G,(G,i),m)),((G,(G,i)) . m)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
(G,F,i,f,m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
f * (G,(G,i),m) is Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,i)), the carrier of F))
diff ((f * (G,(G,i),m)),((G,(G,i)) . m)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
(G,F,i,X,m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
X * (G,(G,i),m) is Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,i)), the carrier of F))
diff ((X * (G,(G,i),m)),((G,(G,i)) . m)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
(G,F,i,f,m) - (G,F,i,X,m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
- (G,F,i,X,m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
(G,F,i,f,m) + (- (G,F,i,X,m)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
the addF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)) is Relation-like K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))) is Relation-like set
K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))) is Relation-like set
K6(K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)))) is set
K462( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)), the addF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)),(G,F,i,f,m),(- (G,F,i,X,m))) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
(f * (G,(G,i),m)) - (X * (G,(G,i),m)) is Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,i)), the carrier of F))
diff (((f * (G,(G,i),m)) - (X * (G,(G,i),m))),((G,(G,i)) . m)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
G is V11() real ext-real Element of REAL
F is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
product F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product F) is non empty non trivial set
dom F is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
f is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of f is non empty non trivial set
K7( the carrier of (product F), the carrier of f) is Relation-like set
K6(K7( the carrier of (product F), the carrier of f)) is set
X is Relation-like the carrier of (product F) -defined the carrier of f -valued Function-like Element of K6(K7( the carrier of (product F), the carrier of f))
G (#) X is Relation-like the carrier of (product F) -defined the carrier of f -valued Function-like Element of K6(K7( the carrier of (product F), the carrier of f))
m is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product F)
i is set
(F,f,i,(G (#) X),m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((F . (F,i)),f))
(F,i) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom F
F . (F,i) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
R_NormSpace_of_BoundedLinearOperators ((F . (F,i)),f) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((F . (F,i)),f) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((F . (F,i)),f)))
R_VectorSpace_of_LinearOperators ((F . (F,i)),f) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((F . (F,i)),f) is non empty functional Element of K6( the carrier of K266( the carrier of (F . (F,i)),f))
the carrier of (F . (F,i)) is non empty non trivial set
K266( the carrier of (F . (F,i)),f) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (F . (F,i)), the carrier of f) is non empty functional FUNCTION_DOMAIN of the carrier of (F . (F,i)), the carrier of f
FuncZero ( the carrier of (F . (F,i)),f) is Relation-like the carrier of (F . (F,i)) -defined the carrier of f -valued Function-like total quasi_total Element of Funcs ( the carrier of (F . (F,i)), the carrier of f)
0. f is V68(f) Element of the carrier of f
the ZeroF of f is Element of the carrier of f
the carrier of (F . (F,i)) --> (0. f) is non empty Relation-like the carrier of (F . (F,i)) -defined the carrier of f -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (F . (F,i)), the carrier of f))
K7( the carrier of (F . (F,i)), the carrier of f) is Relation-like set
K6(K7( the carrier of (F . (F,i)), the carrier of f)) is set
FuncAdd ( the carrier of (F . (F,i)),f) is Relation-like K7((Funcs ( the carrier of (F . (F,i)), the carrier of f)),(Funcs ( the carrier of (F . (F,i)), the carrier of f))) -defined Funcs ( the carrier of (F . (F,i)), the carrier of f) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (F . (F,i)), the carrier of f)),(Funcs ( the carrier of (F . (F,i)), the carrier of f))),(Funcs ( the carrier of (F . (F,i)), the carrier of f))))
K7((Funcs ( the carrier of (F . (F,i)), the carrier of f)),(Funcs ( the carrier of (F . (F,i)), the carrier of f))) is Relation-like set
K7(K7((Funcs ( the carrier of (F . (F,i)), the carrier of f)),(Funcs ( the carrier of (F . (F,i)), the carrier of f))),(Funcs ( the carrier of (F . (F,i)), the carrier of f))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (F . (F,i)), the carrier of f)),(Funcs ( the carrier of (F . (F,i)), the carrier of f))),(Funcs ( the carrier of (F . (F,i)), the carrier of f)))) is set
FuncExtMult ( the carrier of (F . (F,i)),f) is Relation-like K7(REAL,(Funcs ( the carrier of (F . (F,i)), the carrier of f))) -defined Funcs ( the carrier of (F . (F,i)), the carrier of f) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (F . (F,i)), the carrier of f))),(Funcs ( the carrier of (F . (F,i)), the carrier of f))))
K7(REAL,(Funcs ( the carrier of (F . (F,i)), the carrier of f))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (F . (F,i)), the carrier of f))),(Funcs ( the carrier of (F . (F,i)), the carrier of f))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (F . (F,i)), the carrier of f))),(Funcs ( the carrier of (F . (F,i)), the carrier of f)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (F . (F,i)), the carrier of f)),(FuncZero ( the carrier of (F . (F,i)),f)),(FuncAdd ( the carrier of (F . (F,i)),f)),(FuncExtMult ( the carrier of (F . (F,i)),f)) #) is strict RLSStruct
the carrier of K266( the carrier of (F . (F,i)),f) is non empty set
K6( the carrier of K266( the carrier of (F . (F,i)),f)) is set
K215(K266( the carrier of (F . (F,i)),f),(LinearOperators ((F . (F,i)),f))) is Relation-like Function-like Element of LinearOperators ((F . (F,i)),f)
K213(K266( the carrier of (F . (F,i)),f),(LinearOperators ((F . (F,i)),f))) is Relation-like K7((LinearOperators ((F . (F,i)),f)),(LinearOperators ((F . (F,i)),f))) -defined LinearOperators ((F . (F,i)),f) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((F . (F,i)),f)),(LinearOperators ((F . (F,i)),f))),(LinearOperators ((F . (F,i)),f))))
K7((LinearOperators ((F . (F,i)),f)),(LinearOperators ((F . (F,i)),f))) is Relation-like set
K7(K7((LinearOperators ((F . (F,i)),f)),(LinearOperators ((F . (F,i)),f))),(LinearOperators ((F . (F,i)),f))) is Relation-like set
K6(K7(K7((LinearOperators ((F . (F,i)),f)),(LinearOperators ((F . (F,i)),f))),(LinearOperators ((F . (F,i)),f)))) is set
K214(K266( the carrier of (F . (F,i)),f),(LinearOperators ((F . (F,i)),f))) is Relation-like K7(REAL,(LinearOperators ((F . (F,i)),f))) -defined LinearOperators ((F . (F,i)),f) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((F . (F,i)),f))),(LinearOperators ((F . (F,i)),f))))
K7(REAL,(LinearOperators ((F . (F,i)),f))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((F . (F,i)),f))),(LinearOperators ((F . (F,i)),f))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((F . (F,i)),f))),(LinearOperators ((F . (F,i)),f)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((F . (F,i)),f)),K215(K266( the carrier of (F . (F,i)),f),(LinearOperators ((F . (F,i)),f))),K213(K266( the carrier of (F . (F,i)),f),(LinearOperators ((F . (F,i)),f))),K214(K266( the carrier of (F . (F,i)),f),(LinearOperators ((F . (F,i)),f))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((F . (F,i)),f)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((F . (F,i)),f))) is set
K215((R_VectorSpace_of_LinearOperators ((F . (F,i)),f)),(BoundedLinearOperators ((F . (F,i)),f))) is Element of BoundedLinearOperators ((F . (F,i)),f)
K213((R_VectorSpace_of_LinearOperators ((F . (F,i)),f)),(BoundedLinearOperators ((F . (F,i)),f))) is Relation-like K7((BoundedLinearOperators ((F . (F,i)),f)),(BoundedLinearOperators ((F . (F,i)),f))) -defined BoundedLinearOperators ((F . (F,i)),f) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((F . (F,i)),f)),(BoundedLinearOperators ((F . (F,i)),f))),(BoundedLinearOperators ((F . (F,i)),f))))
K7((BoundedLinearOperators ((F . (F,i)),f)),(BoundedLinearOperators ((F . (F,i)),f))) is Relation-like set
K7(K7((BoundedLinearOperators ((F . (F,i)),f)),(BoundedLinearOperators ((F . (F,i)),f))),(BoundedLinearOperators ((F . (F,i)),f))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((F . (F,i)),f)),(BoundedLinearOperators ((F . (F,i)),f))),(BoundedLinearOperators ((F . (F,i)),f)))) is set
K214((R_VectorSpace_of_LinearOperators ((F . (F,i)),f)),(BoundedLinearOperators ((F . (F,i)),f))) is Relation-like K7(REAL,(BoundedLinearOperators ((F . (F,i)),f))) -defined BoundedLinearOperators ((F . (F,i)),f) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((F . (F,i)),f))),(BoundedLinearOperators ((F . (F,i)),f))))
K7(REAL,(BoundedLinearOperators ((F . (F,i)),f))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((F . (F,i)),f))),(BoundedLinearOperators ((F . (F,i)),f))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((F . (F,i)),f))),(BoundedLinearOperators ((F . (F,i)),f)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((F . (F,i)),f) is non empty Relation-like BoundedLinearOperators ((F . (F,i)),f) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((F . (F,i)),f)),REAL))
K7((BoundedLinearOperators ((F . (F,i)),f)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((F . (F,i)),f)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((F . (F,i)),f)),K215((R_VectorSpace_of_LinearOperators ((F . (F,i)),f)),(BoundedLinearOperators ((F . (F,i)),f))),K213((R_VectorSpace_of_LinearOperators ((F . (F,i)),f)),(BoundedLinearOperators ((F . (F,i)),f))),K214((R_VectorSpace_of_LinearOperators ((F . (F,i)),f)),(BoundedLinearOperators ((F . (F,i)),f))),(BoundedLinearOperatorsNorm ((F . (F,i)),f)) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((F . (F,i)),f)) is non empty set
(F,(F,i),m) is non empty Relation-like the carrier of (F . (F,i)) -defined the carrier of (product F) -valued Function-like total quasi_total Element of K6(K7( the carrier of (F . (F,i)), the carrier of (product F)))
K7( the carrier of (F . (F,i)), the carrier of (product F)) is Relation-like set
K6(K7( the carrier of (F . (F,i)), the carrier of (product F))) is set
(G (#) X) * (F,(F,i),m) is Relation-like the carrier of (F . (F,i)) -defined the carrier of f -valued Function-like Element of K6(K7( the carrier of (F . (F,i)), the carrier of f))
(F,(F,i)) is non empty Relation-like the carrier of (product F) -defined the carrier of (F . (F,i)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product F), the carrier of (F . (F,i))))
K7( the carrier of (product F), the carrier of (F . (F,i))) is Relation-like set
K6(K7( the carrier of (product F), the carrier of (F . (F,i)))) is set
(F,(F,i)) . m is Element of the carrier of (F . (F,i))
diff (((G (#) X) * (F,(F,i),m)),((F,(F,i)) . m)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((F . (F,i)),f))
(F,f,i,X,m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((F . (F,i)),f))
X * (F,(F,i),m) is Relation-like the carrier of (F . (F,i)) -defined the carrier of f -valued Function-like Element of K6(K7( the carrier of (F . (F,i)), the carrier of f))
diff ((X * (F,(F,i),m)),((F,(F,i)) . m)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((F . (F,i)),f))
G * (F,f,i,X,m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((F . (F,i)),f))
the Mult of (R_NormSpace_of_BoundedLinearOperators ((F . (F,i)),f)) is Relation-like K7(REAL, the carrier of (R_NormSpace_of_BoundedLinearOperators ((F . (F,i)),f))) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((F . (F,i)),f)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (R_NormSpace_of_BoundedLinearOperators ((F . (F,i)),f))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((F . (F,i)),f))))
K7(REAL, the carrier of (R_NormSpace_of_BoundedLinearOperators ((F . (F,i)),f))) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of (R_NormSpace_of_BoundedLinearOperators ((F . (F,i)),f))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((F . (F,i)),f))) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of (R_NormSpace_of_BoundedLinearOperators ((F . (F,i)),f))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((F . (F,i)),f)))) is non trivial V49() set
K458( the Mult of (R_NormSpace_of_BoundedLinearOperators ((F . (F,i)),f)),G,(F,f,i,X,m)) is set
G (#) (X * (F,(F,i),m)) is Relation-like the carrier of (F . (F,i)) -defined the carrier of f -valued Function-like Element of K6(K7( the carrier of (F . (F,i)), the carrier of f))
diff ((G (#) (X * (F,(F,i),m))),((F,(F,i)) . m)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((F . (F,i)),f))
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . F) is non empty non trivial set
(G,F) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . F) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . F)))
K7( the carrier of (product G), the carrier of (G . F)) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . F))) is set
f is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,F) . f is Element of the carrier of (G . F)
||.((G,F) . f).|| is V11() real ext-real non negative Element of REAL
the normF of (G . F) is non empty Relation-like the carrier of (G . F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (G . F),REAL))
K7( the carrier of (G . F),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (G . F),REAL)) is non trivial V49() set
the normF of (G . F) . ((G,F) . f) is V11() real ext-real Element of REAL
||.f.|| is V11() real ext-real non negative Element of REAL
the normF of (product G) is non empty Relation-like the carrier of (product G) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (product G),REAL))
K7( the carrier of (product G),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (product G),REAL)) is non trivial V49() set
the normF of (product G) . f is V11() real ext-real Element of REAL
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
X is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
(G,X,F) is Element of the carrier of (G . F)
G . F is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . F) is non empty set
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
F is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
dom F is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
dom (carr G) is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
f is set
F . f is set
(carr G) . f is set
len (carr G) is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len G) is non empty V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= len G ) } is set
len F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
K6( the carrier of (product G)) is set
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of F is non empty non trivial set
K7( the carrier of (product G), the carrier of F) is Relation-like set
K6(K7( the carrier of (product G), the carrier of F)) is set
f is set
X is Element of K6( the carrier of (product G))
m is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of F))
dom m is Element of K6( the carrier of (product G))
(G,f) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
G . (G,f) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,f)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,f)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,f)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,f)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,f)),F))
the carrier of (G . (G,f)) is non empty non trivial set
K266( the carrier of (G . (G,f)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,f)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,f)), the carrier of F
FuncZero ( the carrier of (G . (G,f)),F) is Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,f)), the carrier of F)
0. F is V68(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
the carrier of (G . (G,f)) --> (0. F) is non empty Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,f)), the carrier of F))
K7( the carrier of (G . (G,f)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,f)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,f)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,f)), the carrier of F)),(Funcs ( the carrier of (G . (G,f)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,f)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,f)), the carrier of F)),(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,f)), the carrier of F)),(Funcs ( the carrier of (G . (G,f)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,f)), the carrier of F)),(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,f)), the carrier of F)),(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,f)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,f)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,f)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,f)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,f)), the carrier of F)),(FuncZero ( the carrier of (G . (G,f)),F)),(FuncAdd ( the carrier of (G . (G,f)),F)),(FuncExtMult ( the carrier of (G . (G,f)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,f)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,f)),F)) is set
K215(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,f)),F)
K213(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))) is Relation-like K7((LinearOperators ((G . (G,f)),F)),(LinearOperators ((G . (G,f)),F))) -defined LinearOperators ((G . (G,f)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,f)),F)),(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F))))
K7((LinearOperators ((G . (G,f)),F)),(LinearOperators ((G . (G,f)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,f)),F)),(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,f)),F)),(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F)))) is set
K214(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,f)),F))) -defined LinearOperators ((G . (G,f)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F))))
K7(REAL,(LinearOperators ((G . (G,f)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,f)),F)),K215(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))),K213(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))),K214(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,f)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,f)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))) is Element of BoundedLinearOperators ((G . (G,f)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))) -defined BoundedLinearOperators ((G . (G,f)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F))))
K7((BoundedLinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,f)),F))) -defined BoundedLinearOperators ((G . (G,f)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,f)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,f)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,f)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,f)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,f)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,f)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,f)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperatorsNorm ((G . (G,f)),F)) #) is strict NORMSTR
(G,(G,f)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,f)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,f))))
K7( the carrier of (product G), the carrier of (G . (G,f))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,f)))) is set
r is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,(G,f)) . r is Element of the carrier of (G . (G,f))
m | X is Relation-like the carrier of (product G) -defined X -defined the carrier of (product G) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of F))
(G,(G,f),r) is non empty Relation-like the carrier of (G . (G,f)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,f)), the carrier of (product G)))
K7( the carrier of (G . (G,f)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,f)), the carrier of (product G))) is set
(m | X) * (G,(G,f),r) is Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,f)), the carrier of F))
s is Element of the carrier of (G . (G,f))
dom ((m | X) * (G,(G,f),r)) is Element of K6( the carrier of (G . (G,f)))
K6( the carrier of (G . (G,f))) is set
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F)) is non empty set
((m | X) * (G,(G,f),r)) /. s is Element of the carrier of F
y1 is Neighbourhood of s
PP is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F))
DD is Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like RestFunc-like Element of K6(K7( the carrier of (G . (G,f)), the carrier of F))
PP is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F))
DD is Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like RestFunc-like Element of K6(K7( the carrier of (G . (G,f)), the carrier of F))
a is Neighbourhood of s
t is Element of the carrier of (G . (G,f))
(G,(G,f),r) . t is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(dom m) /\ X is Element of K6( the carrier of (product G))
dom (m | X) is Element of K6(X)
K6(X) is set
y1 /\ a is Element of K6( the carrier of (G . (G,f)))
m * (G,(G,f),r) is Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,f)), the carrier of F))
dom (m * (G,(G,f),r)) is Element of K6( the carrier of (G . (G,f)))
t is Neighbourhood of s
dom (G,(G,f),r) is non empty Element of K6( the carrier of (G . (G,f)))
tm is Element of the carrier of (G . (G,f))
(G,(G,f),r) . tm is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,(G,f),r) . s is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((m | X) * (G,(G,f),r)) /. tm is Element of the carrier of F
(G,(G,f),r) /. tm is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(m | X) /. ((G,(G,f),r) /. tm) is Element of the carrier of F
m /. ((G,(G,f),r) /. tm) is Element of the carrier of F
(m * (G,(G,f),r)) /. tm is Element of the carrier of F
(G,(G,f),r) /. s is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(m | X) /. ((G,(G,f),r) /. s) is Element of the carrier of F
m /. ((G,(G,f),r) /. s) is Element of the carrier of F
(m * (G,(G,f),r)) /. s is Element of the carrier of F
((m * (G,(G,f),r)) /. tm) - ((m * (G,(G,f),r)) /. s) is Element of the carrier of F
- ((m * (G,(G,f),r)) /. s) is Element of the carrier of F
((m * (G,(G,f),r)) /. tm) + (- ((m * (G,(G,f),r)) /. s)) is Element of the carrier of F
the addF of F is Relation-like K7( the carrier of F, the carrier of F) -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of F, the carrier of F), the carrier of F))
K7( the carrier of F, the carrier of F) is Relation-like set
K7(K7( the carrier of F, the carrier of F), the carrier of F) is Relation-like set
K6(K7(K7( the carrier of F, the carrier of F), the carrier of F)) is set
K462( the carrier of F, the addF of F,((m * (G,(G,f),r)) /. tm),(- ((m * (G,(G,f),r)) /. s))) is Element of the carrier of F
tm - s is Element of the carrier of (G . (G,f))
- s is Element of the carrier of (G . (G,f))
tm + (- s) is Element of the carrier of (G . (G,f))
the addF of (G . (G,f)) is Relation-like K7( the carrier of (G . (G,f)), the carrier of (G . (G,f))) -defined the carrier of (G . (G,f)) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . (G,f)), the carrier of (G . (G,f))), the carrier of (G . (G,f))))
K7( the carrier of (G . (G,f)), the carrier of (G . (G,f))) is Relation-like set
K7(K7( the carrier of (G . (G,f)), the carrier of (G . (G,f))), the carrier of (G . (G,f))) is Relation-like set
K6(K7(K7( the carrier of (G . (G,f)), the carrier of (G . (G,f))), the carrier of (G . (G,f)))) is set
K462( the carrier of (G . (G,f)), the addF of (G . (G,f)),tm,(- s)) is Element of the carrier of (G . (G,f))
PP . (tm - s) is Element of the carrier of F
DD /. (tm - s) is Element of the carrier of F
(PP . (tm - s)) + (DD /. (tm - s)) is Element of the carrier of F
K462( the carrier of F, the addF of F,(PP . (tm - s)),(DD /. (tm - s))) is Element of the carrier of F
r is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,(G,f)) . r is Element of the carrier of (G . (G,f))
(G,(G,f),r) is non empty Relation-like the carrier of (G . (G,f)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,f)), the carrier of (product G)))
m * (G,(G,f),r) is Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,f)), the carrier of F))
s is Element of the carrier of (G . (G,f))
dom (m * (G,(G,f),r)) is Element of K6( the carrier of (G . (G,f)))
(m * (G,(G,f),r)) /. s is Element of the carrier of F
y1 is Neighbourhood of s
PP is Neighbourhood of s
DD is Element of the carrier of (G . (G,f))
(G,(G,f),r) . DD is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(m | X) * (G,(G,f),r) is Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,f)), the carrier of F))
dom ((m | X) * (G,(G,f),r)) is Element of K6( the carrier of (G . (G,f)))
DD is set
dom (G,(G,f),r) is non empty Element of K6( the carrier of (G . (G,f)))
a is Element of the carrier of (G . (G,f))
(G,(G,f),r) . a is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
y1 /\ PP is Element of K6( the carrier of (G . (G,f)))
DD is Neighbourhood of s
a is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F))
t is Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like RestFunc-like Element of K6(K7( the carrier of (G . (G,f)), the carrier of F))
a is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F))
t is Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like RestFunc-like Element of K6(K7( the carrier of (G . (G,f)), the carrier of F))
tm is Element of the carrier of (G . (G,f))
dom (G,(G,f),r) is non empty Element of K6( the carrier of (G . (G,f)))
(G,(G,f),r) . tm is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,(G,f),r) . s is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((m | X) * (G,(G,f),r)) /. tm is Element of the carrier of F
(G,(G,f),r) /. tm is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(m | X) /. ((G,(G,f),r) /. tm) is Element of the carrier of F
m /. ((G,(G,f),r) /. tm) is Element of the carrier of F
(m * (G,(G,f),r)) /. tm is Element of the carrier of F
((m | X) * (G,(G,f),r)) /. s is Element of the carrier of F
(G,(G,f),r) /. s is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(m | X) /. ((G,(G,f),r) /. s) is Element of the carrier of F
m /. ((G,(G,f),r) /. s) is Element of the carrier of F
(((m | X) * (G,(G,f),r)) /. tm) - (((m | X) * (G,(G,f),r)) /. s) is Element of the carrier of F
- (((m | X) * (G,(G,f),r)) /. s) is Element of the carrier of F
(((m | X) * (G,(G,f),r)) /. tm) + (- (((m | X) * (G,(G,f),r)) /. s)) is Element of the carrier of F
K462( the carrier of F, the addF of F,(((m | X) * (G,(G,f),r)) /. tm),(- (((m | X) * (G,(G,f),r)) /. s))) is Element of the carrier of F
tm - s is Element of the carrier of (G . (G,f))
- s is Element of the carrier of (G . (G,f))
tm + (- s) is Element of the carrier of (G . (G,f))
K462( the carrier of (G . (G,f)), the addF of (G . (G,f)),tm,(- s)) is Element of the carrier of (G . (G,f))
a . (tm - s) is Element of the carrier of F
t /. (tm - s) is Element of the carrier of F
(a . (tm - s)) + (t /. (tm - s)) is Element of the carrier of F
K462( the carrier of F, the addF of F,(a . (tm - s)),(t /. (tm - s))) is Element of the carrier of F
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
0. (product G) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like V68( product G) countable Element of the carrier of (product G)
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
the ZeroF of (product G) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . F) is non empty non trivial set
(G,F,(0. (product G))) is non empty Relation-like the carrier of (G . F) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . F), the carrier of (product G)))
K7( the carrier of (G . F), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . F), the carrier of (product G))) is set
f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . f is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
0. (G . f) is V68(G . f) Element of the carrier of (G . f)
the carrier of (G . f) is non empty non trivial set
the ZeroF of (G . f) is Element of the carrier of (G . f)
X is Element of the carrier of (G . F)
(G,F,(0. (product G))) . X is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
m is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
(G,m,f) is Element of the carrier of (G . f)
G . f is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . f) is non empty set
Replace (F,X,) is Relation-like Function-like set
F .--> X is trivial Relation-like dom G -defined {F} -defined the carrier of (G . F) -valued Function-like one-to-one constant V49() countable set
{F} is non empty trivial V49() V53() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded set
i is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
dom (carr G) is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
r is Relation-like Function-like set
dom r is set
dom i is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below Element of K6(NAT)
(G,i,f) is Element of the carrier of (G . f)
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
0. (product G) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like V68( product G) countable Element of the carrier of (product G)
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
the ZeroF of (product G) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . F) is non empty non trivial set
(G,F,(0. (product G))) is non empty Relation-like the carrier of (G . F) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . F), the carrier of (product G)))
K7( the carrier of (G . F), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . F), the carrier of (product G))) is set
f is Element of the carrier of (G . F)
X is Element of the carrier of (G . F)
f + X is Element of the carrier of (G . F)
the addF of (G . F) is Relation-like K7( the carrier of (G . F), the carrier of (G . F)) -defined the carrier of (G . F) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . F), the carrier of (G . F)), the carrier of (G . F)))
K7( the carrier of (G . F), the carrier of (G . F)) is Relation-like set
K7(K7( the carrier of (G . F), the carrier of (G . F)), the carrier of (G . F)) is Relation-like set
K6(K7(K7( the carrier of (G . F), the carrier of (G . F)), the carrier of (G . F))) is set
K462( the carrier of (G . F), the addF of (G . F),f,X) is Element of the carrier of (G . F)
(G,F,(0. (product G))) . (f + X) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,F,(0. (product G))) . f is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,F,(0. (product G))) . X is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((G,F,(0. (product G))) . f) + ((G,F,(0. (product G))) . X) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
the addF of (product G) is Relation-like K7( the carrier of (product G), the carrier of (product G)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)))
K7( the carrier of (product G), the carrier of (product G)) is Relation-like set
K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)) is Relation-like set
K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G))) is set
K462( the carrier of (product G), the addF of (product G),((G,F,(0. (product G))) . f),((G,F,(0. (product G))) . X)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
m is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
i is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
i0 is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
y0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
(G,m,y0) is Element of the carrier of (G . y0)
G . y0 is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . y0) is non empty set
(G,i,y0) is Element of the carrier of (G . y0)
(G,i0,y0) is Element of the carrier of (G . y0)
(G,i,y0) + (G,i0,y0) is Element of the carrier of (G . y0)
the addF of (G . y0) is Relation-like K7( the carrier of (G . y0), the carrier of (G . y0)) -defined the carrier of (G . y0) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . y0), the carrier of (G . y0)), the carrier of (G . y0)))
K7( the carrier of (G . y0), the carrier of (G . y0)) is Relation-like set
K7(K7( the carrier of (G . y0), the carrier of (G . y0)), the carrier of (G . y0)) is Relation-like set
K6(K7(K7( the carrier of (G . y0), the carrier of (G . y0)), the carrier of (G . y0))) is set
K462( the carrier of (G . y0), the addF of (G . y0),(G,i,y0),(G,i0,y0)) is Element of the carrier of (G . y0)
G . y0 is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . y0) is non empty non trivial set
r is Element of the carrier of (G . y0)
(G,i,y0) + r is Element of the carrier of (G . y0)
the addF of (G . y0) is Relation-like K7( the carrier of (G . y0), the carrier of (G . y0)) -defined the carrier of (G . y0) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . y0), the carrier of (G . y0)), the carrier of (G . y0)))
K7( the carrier of (G . y0), the carrier of (G . y0)) is Relation-like set
K7(K7( the carrier of (G . y0), the carrier of (G . y0)), the carrier of (G . y0)) is Relation-like set
K6(K7(K7( the carrier of (G . y0), the carrier of (G . y0)), the carrier of (G . y0))) is set
K462( the carrier of (G . y0), the addF of (G . y0),(G,i,y0),r) is Element of the carrier of (G . y0)
G . y0 is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
0. (G . y0) is V68(G . y0) Element of the carrier of (G . y0)
the carrier of (G . y0) is non empty non trivial set
the ZeroF of (G . y0) is Element of the carrier of (G . y0)
(0. (G . y0)) + (0. (G . y0)) is Element of the carrier of (G . y0)
the addF of (G . y0) is Relation-like K7( the carrier of (G . y0), the carrier of (G . y0)) -defined the carrier of (G . y0) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . y0), the carrier of (G . y0)), the carrier of (G . y0)))
K7( the carrier of (G . y0), the carrier of (G . y0)) is Relation-like set
K7(K7( the carrier of (G . y0), the carrier of (G . y0)), the carrier of (G . y0)) is Relation-like set
K6(K7(K7( the carrier of (G . y0), the carrier of (G . y0)), the carrier of (G . y0))) is set
K462( the carrier of (G . y0), the addF of (G . y0),(0. (G . y0)),(0. (G . y0))) is Element of the carrier of (G . y0)
(G,i,y0) + (0. (G . y0)) is Element of the carrier of (G . y0)
K462( the carrier of (G . y0), the addF of (G . y0),(G,i,y0),(0. (G . y0))) is Element of the carrier of (G . y0)
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . F) is non empty non trivial set
(G,F) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . F) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . F)))
K7( the carrier of (product G), the carrier of (G . F)) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . F))) is set
f is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
X is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f + X is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
the addF of (product G) is Relation-like K7( the carrier of (product G), the carrier of (product G)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)))
K7( the carrier of (product G), the carrier of (product G)) is Relation-like set
K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)) is Relation-like set
K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G))) is set
K462( the carrier of (product G), the addF of (product G),f,X) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,F) . (f + X) is Element of the carrier of (G . F)
(G,F) . f is Element of the carrier of (G . F)
(G,F) . X is Element of the carrier of (G . F)
((G,F) . f) + ((G,F) . X) is Element of the carrier of (G . F)
the addF of (G . F) is Relation-like K7( the carrier of (G . F), the carrier of (G . F)) -defined the carrier of (G . F) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . F), the carrier of (G . F)), the carrier of (G . F)))
K7( the carrier of (G . F), the carrier of (G . F)) is Relation-like set
K7(K7( the carrier of (G . F), the carrier of (G . F)), the carrier of (G . F)) is Relation-like set
K6(K7(K7( the carrier of (G . F), the carrier of (G . F)), the carrier of (G . F))) is set
K462( the carrier of (G . F), the addF of (G . F),((G,F) . f),((G,F) . X)) is Element of the carrier of (G . F)
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
m is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
(G,m,F) is Element of the carrier of (G . F)
G . F is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . F) is non empty set
i is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
(G,i,F) is Element of the carrier of (G . F)
i0 is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
(G,i0,F) is Element of the carrier of (G . F)
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
0. (product G) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like V68( product G) countable Element of the carrier of (product G)
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
the ZeroF of (product G) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . F) is non empty non trivial set
(G,F,(0. (product G))) is non empty Relation-like the carrier of (G . F) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . F), the carrier of (product G)))
K7( the carrier of (G . F), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . F), the carrier of (product G))) is set
f is Element of the carrier of (G . F)
X is Element of the carrier of (G . F)
f - X is Element of the carrier of (G . F)
- X is Element of the carrier of (G . F)
f + (- X) is Element of the carrier of (G . F)
the addF of (G . F) is Relation-like K7( the carrier of (G . F), the carrier of (G . F)) -defined the carrier of (G . F) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . F), the carrier of (G . F)), the carrier of (G . F)))
K7( the carrier of (G . F), the carrier of (G . F)) is Relation-like set
K7(K7( the carrier of (G . F), the carrier of (G . F)), the carrier of (G . F)) is Relation-like set
K6(K7(K7( the carrier of (G . F), the carrier of (G . F)), the carrier of (G . F))) is set
K462( the carrier of (G . F), the addF of (G . F),f,(- X)) is Element of the carrier of (G . F)
(G,F,(0. (product G))) . (f - X) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,F,(0. (product G))) . f is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,F,(0. (product G))) . X is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((G,F,(0. (product G))) . f) - ((G,F,(0. (product G))) . X) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
- ((G,F,(0. (product G))) . X) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((G,F,(0. (product G))) . f) + (- ((G,F,(0. (product G))) . X)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
the addF of (product G) is Relation-like K7( the carrier of (product G), the carrier of (product G)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)))
K7( the carrier of (product G), the carrier of (product G)) is Relation-like set
K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)) is Relation-like set
K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G))) is set
K462( the carrier of (product G), the addF of (product G),((G,F,(0. (product G))) . f),(- ((G,F,(0. (product G))) . X))) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
m is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
i is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
i0 is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
y0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
(G,m,y0) is Element of the carrier of (G . y0)
G . y0 is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . y0) is non empty set
(G,i,y0) is Element of the carrier of (G . y0)
(G,i0,y0) is Element of the carrier of (G . y0)
(G,i,y0) - (G,i0,y0) is Element of the carrier of (G . y0)
- (G,i0,y0) is Element of the carrier of (G . y0)
(G,i,y0) + (- (G,i0,y0)) is Element of the carrier of (G . y0)
the addF of (G . y0) is Relation-like K7( the carrier of (G . y0), the carrier of (G . y0)) -defined the carrier of (G . y0) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . y0), the carrier of (G . y0)), the carrier of (G . y0)))
K7( the carrier of (G . y0), the carrier of (G . y0)) is Relation-like set
K7(K7( the carrier of (G . y0), the carrier of (G . y0)), the carrier of (G . y0)) is Relation-like set
K6(K7(K7( the carrier of (G . y0), the carrier of (G . y0)), the carrier of (G . y0))) is set
K462( the carrier of (G . y0), the addF of (G . y0),(G,i,y0),(- (G,i0,y0))) is Element of the carrier of (G . y0)
G . y0 is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . y0) is non empty non trivial set
r is Element of the carrier of (G . y0)
(G,i,y0) - r is Element of the carrier of (G . y0)
- r is Element of the carrier of (G . y0)
(G,i,y0) + (- r) is Element of the carrier of (G . y0)
the addF of (G . y0) is Relation-like K7( the carrier of (G . y0), the carrier of (G . y0)) -defined the carrier of (G . y0) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . y0), the carrier of (G . y0)), the carrier of (G . y0)))
K7( the carrier of (G . y0), the carrier of (G . y0)) is Relation-like set
K7(K7( the carrier of (G . y0), the carrier of (G . y0)), the carrier of (G . y0)) is Relation-like set
K6(K7(K7( the carrier of (G . y0), the carrier of (G . y0)), the carrier of (G . y0))) is set
K462( the carrier of (G . y0), the addF of (G . y0),(G,i,y0),(- r)) is Element of the carrier of (G . y0)
G . y0 is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
0. (G . y0) is V68(G . y0) Element of the carrier of (G . y0)
the carrier of (G . y0) is non empty non trivial set
the ZeroF of (G . y0) is Element of the carrier of (G . y0)
(0. (G . y0)) - (0. (G . y0)) is Element of the carrier of (G . y0)
- (0. (G . y0)) is Element of the carrier of (G . y0)
(0. (G . y0)) + (- (0. (G . y0))) is Element of the carrier of (G . y0)
the addF of (G . y0) is Relation-like K7( the carrier of (G . y0), the carrier of (G . y0)) -defined the carrier of (G . y0) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . y0), the carrier of (G . y0)), the carrier of (G . y0)))
K7( the carrier of (G . y0), the carrier of (G . y0)) is Relation-like set
K7(K7( the carrier of (G . y0), the carrier of (G . y0)), the carrier of (G . y0)) is Relation-like set
K6(K7(K7( the carrier of (G . y0), the carrier of (G . y0)), the carrier of (G . y0))) is set
K462( the carrier of (G . y0), the addF of (G . y0),(0. (G . y0)),(- (0. (G . y0)))) is Element of the carrier of (G . y0)
(G,i,y0) - (0. (G . y0)) is Element of the carrier of (G . y0)
(G,i,y0) + (- (0. (G . y0))) is Element of the carrier of (G . y0)
K462( the carrier of (G . y0), the addF of (G . y0),(G,i,y0),(- (0. (G . y0)))) is Element of the carrier of (G . y0)
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . F) is non empty non trivial set
(G,F) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . F) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . F)))
K7( the carrier of (product G), the carrier of (G . F)) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . F))) is set
f is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
X is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f - X is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
- X is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f + (- X) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
the addF of (product G) is Relation-like K7( the carrier of (product G), the carrier of (product G)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)))
K7( the carrier of (product G), the carrier of (product G)) is Relation-like set
K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)) is Relation-like set
K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G))) is set
K462( the carrier of (product G), the addF of (product G),f,(- X)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,F) . (f - X) is Element of the carrier of (G . F)
(G,F) . f is Element of the carrier of (G . F)
(G,F) . X is Element of the carrier of (G . F)
((G,F) . f) - ((G,F) . X) is Element of the carrier of (G . F)
- ((G,F) . X) is Element of the carrier of (G . F)
((G,F) . f) + (- ((G,F) . X)) is Element of the carrier of (G . F)
the addF of (G . F) is Relation-like K7( the carrier of (G . F), the carrier of (G . F)) -defined the carrier of (G . F) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . F), the carrier of (G . F)), the carrier of (G . F)))
K7( the carrier of (G . F), the carrier of (G . F)) is Relation-like set
K7(K7( the carrier of (G . F), the carrier of (G . F)), the carrier of (G . F)) is Relation-like set
K6(K7(K7( the carrier of (G . F), the carrier of (G . F)), the carrier of (G . F))) is set
K462( the carrier of (G . F), the addF of (G . F),((G,F) . f),(- ((G,F) . X))) is Element of the carrier of (G . F)
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
m is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
(G,m,F) is Element of the carrier of (G . F)
G . F is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . F) is non empty set
i is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
(G,i,F) is Element of the carrier of (G . F)
i0 is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
(G,i0,F) is Element of the carrier of (G . F)
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
0. (product G) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like V68( product G) countable Element of the carrier of (product G)
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
the ZeroF of (product G) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . F) is non empty non trivial set
0. (G . F) is V68(G . F) Element of the carrier of (G . F)
the ZeroF of (G . F) is Element of the carrier of (G . F)
(G,F,(0. (product G))) is non empty Relation-like the carrier of (G . F) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . F), the carrier of (product G)))
K7( the carrier of (G . F), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . F), the carrier of (product G))) is set
f is Element of the carrier of (G . F)
(G,F,(0. (product G))) . f is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
X is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
(G,X,F) is Element of the carrier of (G . F)
G . F is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . F) is non empty set
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
0. (product G) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like V68( product G) countable Element of the carrier of (product G)
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
the ZeroF of (product G) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . F) is non empty non trivial set
(G,F,(0. (product G))) is non empty Relation-like the carrier of (G . F) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . F), the carrier of (product G)))
K7( the carrier of (G . F), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . F), the carrier of (product G))) is set
f is Element of the carrier of (G . F)
(G,F,(0. (product G))) . f is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
X is V11() real ext-real Element of REAL
X * f is Element of the carrier of (G . F)
the Mult of (G . F) is Relation-like K7(REAL, the carrier of (G . F)) -defined the carrier of (G . F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (G . F)), the carrier of (G . F)))
K7(REAL, the carrier of (G . F)) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of (G . F)), the carrier of (G . F)) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of (G . F)), the carrier of (G . F))) is non trivial V49() set
K458( the Mult of (G . F),X,f) is set
(G,F,(0. (product G))) . (X * f) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
X * ((G,F,(0. (product G))) . f) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
the Mult of (product G) is Relation-like K7(REAL, the carrier of (product G)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (product G)), the carrier of (product G)))
K7(REAL, the carrier of (product G)) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of (product G)), the carrier of (product G)) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of (product G)), the carrier of (product G))) is non trivial V49() set
K458( the Mult of (product G),X,((G,F,(0. (product G))) . f)) is set
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
m is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
i is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
i0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
(G,m,i0) is Element of the carrier of (G . i0)
G . i0 is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . i0) is non empty set
(G,i,i0) is Element of the carrier of (G . i0)
X * (G,i,i0) is Element of the carrier of (G . i0)
the Mult of (G . i0) is Relation-like K7(REAL, the carrier of (G . i0)) -defined the carrier of (G . i0) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (G . i0)), the carrier of (G . i0)))
K7(REAL, the carrier of (G . i0)) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of (G . i0)), the carrier of (G . i0)) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of (G . i0)), the carrier of (G . i0))) is non trivial V49() set
K458( the Mult of (G . i0),X,(G,i,i0)) is set
G . i0 is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
0. (G . i0) is V68(G . i0) Element of the carrier of (G . i0)
the carrier of (G . i0) is non empty non trivial set
the ZeroF of (G . i0) is Element of the carrier of (G . i0)
X * (0. (G . i0)) is Element of the carrier of (G . i0)
the Mult of (G . i0) is Relation-like K7(REAL, the carrier of (G . i0)) -defined the carrier of (G . i0) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (G . i0)), the carrier of (G . i0)))
K7(REAL, the carrier of (G . i0)) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of (G . i0)), the carrier of (G . i0)) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of (G . i0)), the carrier of (G . i0))) is non trivial V49() set
K458( the Mult of (G . i0),X,(0. (G . i0))) is set
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . F) is non empty non trivial set
(G,F) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . F) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . F)))
K7( the carrier of (product G), the carrier of (G . F)) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . F))) is set
f is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,F) . f is Element of the carrier of (G . F)
X is V11() real ext-real Element of REAL
X * f is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
the Mult of (product G) is Relation-like K7(REAL, the carrier of (product G)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (product G)), the carrier of (product G)))
K7(REAL, the carrier of (product G)) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of (product G)), the carrier of (product G)) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of (product G)), the carrier of (product G))) is non trivial V49() set
K458( the Mult of (product G),X,f) is set
(G,F) . (X * f) is Element of the carrier of (G . F)
X * ((G,F) . f) is Element of the carrier of (G . F)
the Mult of (G . F) is Relation-like K7(REAL, the carrier of (G . F)) -defined the carrier of (G . F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (G . F)), the carrier of (G . F)))
K7(REAL, the carrier of (G . F)) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of (G . F)), the carrier of (G . F)) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of (G . F)), the carrier of (G . F))) is non trivial V49() set
K458( the Mult of (G . F),X,((G,F) . f)) is set
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
m is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
(G,m,F) is Element of the carrier of (G . F)
G . F is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . F) is non empty set
i is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
(G,i,F) is Element of the carrier of (G . F)
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
0. (product G) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like V68( product G) countable Element of the carrier of (product G)
the ZeroF of (product G) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of F is non empty non trivial set
K7( the carrier of (product G), the carrier of F) is Relation-like set
K6(K7( the carrier of (product G), the carrier of F)) is set
f is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of F))
X is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
diff (f,X) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
R_NormSpace_of_BoundedLinearOperators ((product G),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((product G),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((product G),F)))
R_VectorSpace_of_LinearOperators ((product G),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((product G),F) is non empty functional Element of K6( the carrier of K266( the carrier of (product G),F))
K266( the carrier of (product G),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (product G), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (product G), the carrier of F
FuncZero ( the carrier of (product G),F) is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (product G), the carrier of F)
0. F is V68(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
the carrier of (product G) --> (0. F) is non empty Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (product G), the carrier of F))
FuncAdd ( the carrier of (product G),F) is Relation-like K7((Funcs ( the carrier of (product G), the carrier of F)),(Funcs ( the carrier of (product G), the carrier of F))) -defined Funcs ( the carrier of (product G), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (product G), the carrier of F)),(Funcs ( the carrier of (product G), the carrier of F))),(Funcs ( the carrier of (product G), the carrier of F))))
K7((Funcs ( the carrier of (product G), the carrier of F)),(Funcs ( the carrier of (product G), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (product G), the carrier of F)),(Funcs ( the carrier of (product G), the carrier of F))),(Funcs ( the carrier of (product G), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (product G), the carrier of F)),(Funcs ( the carrier of (product G), the carrier of F))),(Funcs ( the carrier of (product G), the carrier of F)))) is set
FuncExtMult ( the carrier of (product G),F) is Relation-like K7(REAL,(Funcs ( the carrier of (product G), the carrier of F))) -defined Funcs ( the carrier of (product G), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (product G), the carrier of F))),(Funcs ( the carrier of (product G), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (product G), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (product G), the carrier of F))),(Funcs ( the carrier of (product G), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (product G), the carrier of F))),(Funcs ( the carrier of (product G), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (product G), the carrier of F)),(FuncZero ( the carrier of (product G),F)),(FuncAdd ( the carrier of (product G),F)),(FuncExtMult ( the carrier of (product G),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (product G),F) is non empty set
K6( the carrier of K266( the carrier of (product G),F)) is set
K215(K266( the carrier of (product G),F),(LinearOperators ((product G),F))) is Relation-like Function-like Element of LinearOperators ((product G),F)
K213(K266( the carrier of (product G),F),(LinearOperators ((product G),F))) is Relation-like K7((LinearOperators ((product G),F)),(LinearOperators ((product G),F))) -defined LinearOperators ((product G),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((product G),F)),(LinearOperators ((product G),F))),(LinearOperators ((product G),F))))
K7((LinearOperators ((product G),F)),(LinearOperators ((product G),F))) is Relation-like set
K7(K7((LinearOperators ((product G),F)),(LinearOperators ((product G),F))),(LinearOperators ((product G),F))) is Relation-like set
K6(K7(K7((LinearOperators ((product G),F)),(LinearOperators ((product G),F))),(LinearOperators ((product G),F)))) is set
K214(K266( the carrier of (product G),F),(LinearOperators ((product G),F))) is Relation-like K7(REAL,(LinearOperators ((product G),F))) -defined LinearOperators ((product G),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((product G),F))),(LinearOperators ((product G),F))))
K7(REAL,(LinearOperators ((product G),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((product G),F))),(LinearOperators ((product G),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((product G),F))),(LinearOperators ((product G),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((product G),F)),K215(K266( the carrier of (product G),F),(LinearOperators ((product G),F))),K213(K266( the carrier of (product G),F),(LinearOperators ((product G),F))),K214(K266( the carrier of (product G),F),(LinearOperators ((product G),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((product G),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((product G),F))) is set
K215((R_VectorSpace_of_LinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))) is Element of BoundedLinearOperators ((product G),F)
K213((R_VectorSpace_of_LinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))) is Relation-like K7((BoundedLinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))) -defined BoundedLinearOperators ((product G),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))),(BoundedLinearOperators ((product G),F))))
K7((BoundedLinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))),(BoundedLinearOperators ((product G),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))),(BoundedLinearOperators ((product G),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((product G),F))) -defined BoundedLinearOperators ((product G),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((product G),F))),(BoundedLinearOperators ((product G),F))))
K7(REAL,(BoundedLinearOperators ((product G),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((product G),F))),(BoundedLinearOperators ((product G),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((product G),F))),(BoundedLinearOperators ((product G),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((product G),F) is non empty Relation-like BoundedLinearOperators ((product G),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((product G),F)),REAL))
K7((BoundedLinearOperators ((product G),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((product G),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((product G),F)),K215((R_VectorSpace_of_LinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))),K213((R_VectorSpace_of_LinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))),K214((R_VectorSpace_of_LinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))),(BoundedLinearOperatorsNorm ((product G),F)) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)) is non empty set
m is set
(G,F,m,f,X) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F))
(G,m) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
G . (G,m) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,m)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,m)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,m)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,m)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,m)),F))
the carrier of (G . (G,m)) is non empty non trivial set
K266( the carrier of (G . (G,m)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,m)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,m)), the carrier of F
FuncZero ( the carrier of (G . (G,m)),F) is Relation-like the carrier of (G . (G,m)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,m)), the carrier of F)
the carrier of (G . (G,m)) --> (0. F) is non empty Relation-like the carrier of (G . (G,m)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,m)), the carrier of F))
K7( the carrier of (G . (G,m)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,m)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,m)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,m)), the carrier of F)),(Funcs ( the carrier of (G . (G,m)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,m)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,m)), the carrier of F)),(Funcs ( the carrier of (G . (G,m)), the carrier of F))),(Funcs ( the carrier of (G . (G,m)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,m)), the carrier of F)),(Funcs ( the carrier of (G . (G,m)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,m)), the carrier of F)),(Funcs ( the carrier of (G . (G,m)), the carrier of F))),(Funcs ( the carrier of (G . (G,m)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,m)), the carrier of F)),(Funcs ( the carrier of (G . (G,m)), the carrier of F))),(Funcs ( the carrier of (G . (G,m)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,m)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,m)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,m)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,m)), the carrier of F))),(Funcs ( the carrier of (G . (G,m)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,m)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,m)), the carrier of F))),(Funcs ( the carrier of (G . (G,m)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,m)), the carrier of F))),(Funcs ( the carrier of (G . (G,m)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,m)), the carrier of F)),(FuncZero ( the carrier of (G . (G,m)),F)),(FuncAdd ( the carrier of (G . (G,m)),F)),(FuncExtMult ( the carrier of (G . (G,m)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,m)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,m)),F)) is set
K215(K266( the carrier of (G . (G,m)),F),(LinearOperators ((G . (G,m)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,m)),F)
K213(K266( the carrier of (G . (G,m)),F),(LinearOperators ((G . (G,m)),F))) is Relation-like K7((LinearOperators ((G . (G,m)),F)),(LinearOperators ((G . (G,m)),F))) -defined LinearOperators ((G . (G,m)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,m)),F)),(LinearOperators ((G . (G,m)),F))),(LinearOperators ((G . (G,m)),F))))
K7((LinearOperators ((G . (G,m)),F)),(LinearOperators ((G . (G,m)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,m)),F)),(LinearOperators ((G . (G,m)),F))),(LinearOperators ((G . (G,m)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,m)),F)),(LinearOperators ((G . (G,m)),F))),(LinearOperators ((G . (G,m)),F)))) is set
K214(K266( the carrier of (G . (G,m)),F),(LinearOperators ((G . (G,m)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,m)),F))) -defined LinearOperators ((G . (G,m)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,m)),F))),(LinearOperators ((G . (G,m)),F))))
K7(REAL,(LinearOperators ((G . (G,m)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,m)),F))),(LinearOperators ((G . (G,m)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,m)),F))),(LinearOperators ((G . (G,m)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,m)),F)),K215(K266( the carrier of (G . (G,m)),F),(LinearOperators ((G . (G,m)),F))),K213(K266( the carrier of (G . (G,m)),F),(LinearOperators ((G . (G,m)),F))),K214(K266( the carrier of (G . (G,m)),F),(LinearOperators ((G . (G,m)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,m)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,m)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,m)),F)),(BoundedLinearOperators ((G . (G,m)),F))) is Element of BoundedLinearOperators ((G . (G,m)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,m)),F)),(BoundedLinearOperators ((G . (G,m)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,m)),F)),(BoundedLinearOperators ((G . (G,m)),F))) -defined BoundedLinearOperators ((G . (G,m)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,m)),F)),(BoundedLinearOperators ((G . (G,m)),F))),(BoundedLinearOperators ((G . (G,m)),F))))
K7((BoundedLinearOperators ((G . (G,m)),F)),(BoundedLinearOperators ((G . (G,m)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,m)),F)),(BoundedLinearOperators ((G . (G,m)),F))),(BoundedLinearOperators ((G . (G,m)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,m)),F)),(BoundedLinearOperators ((G . (G,m)),F))),(BoundedLinearOperators ((G . (G,m)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,m)),F)),(BoundedLinearOperators ((G . (G,m)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,m)),F))) -defined BoundedLinearOperators ((G . (G,m)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,m)),F))),(BoundedLinearOperators ((G . (G,m)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,m)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,m)),F))),(BoundedLinearOperators ((G . (G,m)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,m)),F))),(BoundedLinearOperators ((G . (G,m)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,m)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,m)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,m)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,m)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,m)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,m)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,m)),F)),(BoundedLinearOperators ((G . (G,m)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,m)),F)),(BoundedLinearOperators ((G . (G,m)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,m)),F)),(BoundedLinearOperators ((G . (G,m)),F))),(BoundedLinearOperatorsNorm ((G . (G,m)),F)) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F)) is non empty set
(G,(G,m),X) is non empty Relation-like the carrier of (G . (G,m)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,m)), the carrier of (product G)))
K7( the carrier of (G . (G,m)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,m)), the carrier of (product G))) is set
f * (G,(G,m),X) is Relation-like the carrier of (G . (G,m)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,m)), the carrier of F))
(G,(G,m)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,m)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,m))))
K7( the carrier of (product G), the carrier of (G . (G,m))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,m)))) is set
(G,(G,m)) . X is Element of the carrier of (G . (G,m))
diff ((f * (G,(G,m),X)),((G,(G,m)) . X)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F))
(G,(G,m),(0. (product G))) is non empty Relation-like the carrier of (G . (G,m)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,m)), the carrier of (product G)))
(G,(G,m),(0. (product G))) (#) (diff (f,X)) is Relation-like the carrier of (G . (G,m)) -defined Function-like set
dom f is Element of K6( the carrier of (product G))
K6( the carrier of (product G)) is set
f /. X is Element of the carrier of F
i0 is Neighbourhood of X
y0 is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like RestFunc-like Element of K6(K7( the carrier of (product G), the carrier of F))
y0 is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like RestFunc-like Element of K6(K7( the carrier of (product G), the carrier of F))
r is V11() real ext-real Element of REAL
{ b₁ where b₁ is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G) : not r <= ||.(b₁ - X).|| } is set
y1 is Element of the carrier of (G . (G,m))
{ b₁ where b₁ is Element of the carrier of (G . (G,m)) : not r <= ||.(b₁ - y1).|| } is set
a is set
t is Element of the carrier of (G . (G,m))
t - y1 is Element of the carrier of (G . (G,m))
- y1 is Element of the carrier of (G . (G,m))
t + (- y1) is Element of the carrier of (G . (G,m))
the addF of (G . (G,m)) is Relation-like K7( the carrier of (G . (G,m)), the carrier of (G . (G,m))) -defined the carrier of (G . (G,m)) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . (G,m)), the carrier of (G . (G,m))), the carrier of (G . (G,m))))
K7( the carrier of (G . (G,m)), the carrier of (G . (G,m))) is Relation-like set
K7(K7( the carrier of (G . (G,m)), the carrier of (G . (G,m))), the carrier of (G . (G,m))) is Relation-like set
K6(K7(K7( the carrier of (G . (G,m)), the carrier of (G . (G,m))), the carrier of (G . (G,m)))) is set
K462( the carrier of (G . (G,m)), the addF of (G . (G,m)),t,(- y1)) is Element of the carrier of (G . (G,m))
||.(t - y1).|| is V11() real ext-real non negative Element of REAL
the normF of (G . (G,m)) is non empty Relation-like the carrier of (G . (G,m)) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (G . (G,m)),REAL))
K7( the carrier of (G . (G,m)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (G . (G,m)),REAL)) is non trivial V49() set
the normF of (G . (G,m)) . (t - y1) is V11() real ext-real Element of REAL
K6( the carrier of (G . (G,m))) is set
a is Neighbourhood of y1
t is Element of the carrier of (G . (G,m))
(G,(G,m),X) . t is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((G,(G,m),X) . t) - X is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
- X is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((G,(G,m),X) . t) + (- X) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
the addF of (product G) is Relation-like K7( the carrier of (product G), the carrier of (product G)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)))
K7( the carrier of (product G), the carrier of (product G)) is Relation-like set
K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)) is Relation-like set
K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G))) is set
K462( the carrier of (product G), the addF of (product G),((G,(G,m),X) . t),(- X)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
t - y1 is Element of the carrier of (G . (G,m))
- y1 is Element of the carrier of (G . (G,m))
t + (- y1) is Element of the carrier of (G . (G,m))
the addF of (G . (G,m)) is Relation-like K7( the carrier of (G . (G,m)), the carrier of (G . (G,m))) -defined the carrier of (G . (G,m)) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . (G,m)), the carrier of (G . (G,m))), the carrier of (G . (G,m))))
K7( the carrier of (G . (G,m)), the carrier of (G . (G,m))) is Relation-like set
K7(K7( the carrier of (G . (G,m)), the carrier of (G . (G,m))), the carrier of (G . (G,m))) is Relation-like set
K6(K7(K7( the carrier of (G . (G,m)), the carrier of (G . (G,m))), the carrier of (G . (G,m)))) is set
K462( the carrier of (G . (G,m)), the addF of (G . (G,m)),t,(- y1)) is Element of the carrier of (G . (G,m))
(G,(G,m),(0. (product G))) . (t - y1) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.(((G,(G,m),X) . t) - X).|| is V11() real ext-real non negative Element of REAL
the normF of (product G) is non empty Relation-like the carrier of (product G) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (product G),REAL))
K7( the carrier of (product G),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (product G),REAL)) is non trivial V49() set
the normF of (product G) . (((G,(G,m),X) . t) - X) is V11() real ext-real Element of REAL
tm is Element of the carrier of (G . (G,m))
tm - y1 is Element of the carrier of (G . (G,m))
tm + (- y1) is Element of the carrier of (G . (G,m))
K462( the carrier of (G . (G,m)), the addF of (G . (G,m)),tm,(- y1)) is Element of the carrier of (G . (G,m))
||.(tm - y1).|| is V11() real ext-real non negative Element of REAL
the normF of (G . (G,m)) is non empty Relation-like the carrier of (G . (G,m)) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (G . (G,m)),REAL))
K7( the carrier of (G . (G,m)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (G . (G,m)),REAL)) is non trivial V49() set
the normF of (G . (G,m)) . (tm - y1) is V11() real ext-real Element of REAL
dom y0 is Element of K6( the carrier of (product G))
y0 * (G,(G,m),(0. (product G))) is Relation-like the carrier of (G . (G,m)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,m)), the carrier of F))
dom (G,(G,m),(0. (product G))) is non empty Element of K6( the carrier of (G . (G,m)))
t is Relation-like the carrier of (G . (G,m)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,m)), the carrier of F))
dom t is Element of K6( the carrier of (G . (G,m)))
0. (G . (G,m)) is V68(G . (G,m)) Element of the carrier of (G . (G,m))
the ZeroF of (G . (G,m)) is Element of the carrier of (G . (G,m))
tm is V11() real ext-real Element of REAL
Sw1 is V11() real ext-real Element of REAL
w2 is Element of the carrier of (G . (G,m))
||.w2.|| is V11() real ext-real non negative Element of REAL
the normF of (G . (G,m)) is non empty Relation-like the carrier of (G . (G,m)) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (G . (G,m)),REAL))
K7( the carrier of (G . (G,m)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (G . (G,m)),REAL)) is non trivial V49() set
the normF of (G . (G,m)) . w2 is V11() real ext-real Element of REAL
(G,(G,m),(0. (product G))) . w2 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.((G,(G,m),(0. (product G))) . w2).|| is V11() real ext-real non negative Element of REAL
the normF of (product G) is non empty Relation-like the carrier of (product G) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (product G),REAL))
K7( the carrier of (product G),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (product G),REAL)) is non trivial V49() set
the normF of (product G) . ((G,(G,m),(0. (product G))) . w2) is V11() real ext-real Element of REAL
y0 /. ((G,(G,m),(0. (product G))) . w2) is Element of the carrier of F
y0 . ((G,(G,m),(0. (product G))) . w2) is set
t . w2 is set
t /. w2 is Element of the carrier of F
||.w2.|| " is V11() real ext-real non negative Element of REAL
||.(t /. w2).|| is V11() real ext-real non negative Element of REAL
the normF of F is non empty Relation-like the carrier of F -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of F,REAL))
K7( the carrier of F,REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of F,REAL)) is non trivial V49() set
the normF of F . (t /. w2) is V11() real ext-real Element of REAL
(||.w2.|| ") * ||.(t /. w2).|| is V11() real ext-real non negative Element of REAL
w2 is Element of the carrier of (G . (G,m))
||.w2.|| is V11() real ext-real non negative Element of REAL
the normF of (G . (G,m)) . w2 is V11() real ext-real Element of REAL
||.w2.|| " is V11() real ext-real non negative Element of REAL
t /. w2 is Element of the carrier of F
||.(t /. w2).|| is V11() real ext-real non negative Element of REAL
the normF of F . (t /. w2) is V11() real ext-real Element of REAL
(||.w2.|| ") * ||.(t /. w2).|| is V11() real ext-real non negative Element of REAL
Sw1 is non empty Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like total quasi_total V153( product G,F) V154( product G,F) Lipschitzian Element of K6(K7( the carrier of (product G), the carrier of F))
Sw1 * (G,(G,m),(0. (product G))) is non empty Relation-like the carrier of (G . (G,m)) -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,m)), the carrier of F))
w2 is non empty Relation-like the carrier of (G . (G,m)) -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,m)), the carrier of F))
ys is Element of the carrier of (G . (G,m))
rm is Element of the carrier of (G . (G,m))
ys + rm is Element of the carrier of (G . (G,m))
the addF of (G . (G,m)) is Relation-like K7( the carrier of (G . (G,m)), the carrier of (G . (G,m))) -defined the carrier of (G . (G,m)) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . (G,m)), the carrier of (G . (G,m))), the carrier of (G . (G,m))))
K7( the carrier of (G . (G,m)), the carrier of (G . (G,m))) is Relation-like set
K7(K7( the carrier of (G . (G,m)), the carrier of (G . (G,m))), the carrier of (G . (G,m))) is Relation-like set
K6(K7(K7( the carrier of (G . (G,m)), the carrier of (G . (G,m))), the carrier of (G . (G,m)))) is set
K462( the carrier of (G . (G,m)), the addF of (G . (G,m)),ys,rm) is Element of the carrier of (G . (G,m))
w2 . (ys + rm) is Element of the carrier of F
(G,(G,m),(0. (product G))) . (ys + rm) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
Sw1 . ((G,(G,m),(0. (product G))) . (ys + rm)) is Element of the carrier of F
(G,(G,m),(0. (product G))) . ys is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,(G,m),(0. (product G))) . rm is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((G,(G,m),(0. (product G))) . ys) + ((G,(G,m),(0. (product G))) . rm) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
the addF of (product G) is Relation-like K7( the carrier of (product G), the carrier of (product G)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)))
K7( the carrier of (product G), the carrier of (product G)) is Relation-like set
K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)) is Relation-like set
K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G))) is set
K462( the carrier of (product G), the addF of (product G),((G,(G,m),(0. (product G))) . ys),((G,(G,m),(0. (product G))) . rm)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
Sw1 . (((G,(G,m),(0. (product G))) . ys) + ((G,(G,m),(0. (product G))) . rm)) is Element of the carrier of F
Sw1 . ((G,(G,m),(0. (product G))) . ys) is Element of the carrier of F
Sw1 . ((G,(G,m),(0. (product G))) . rm) is Element of the carrier of F
(Sw1 . ((G,(G,m),(0. (product G))) . ys)) + (Sw1 . ((G,(G,m),(0. (product G))) . rm)) is Element of the carrier of F
the addF of F is Relation-like K7( the carrier of F, the carrier of F) -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of F, the carrier of F), the carrier of F))
K7( the carrier of F, the carrier of F) is Relation-like set
K7(K7( the carrier of F, the carrier of F), the carrier of F) is Relation-like set
K6(K7(K7( the carrier of F, the carrier of F), the carrier of F)) is set
K462( the carrier of F, the addF of F,(Sw1 . ((G,(G,m),(0. (product G))) . ys)),(Sw1 . ((G,(G,m),(0. (product G))) . rm))) is Element of the carrier of F
w2 . ys is Element of the carrier of F
(w2 . ys) + (Sw1 . ((G,(G,m),(0. (product G))) . rm)) is Element of the carrier of F
K462( the carrier of F, the addF of F,(w2 . ys),(Sw1 . ((G,(G,m),(0. (product G))) . rm))) is Element of the carrier of F
w2 . rm is Element of the carrier of F
(w2 . ys) + (w2 . rm) is Element of the carrier of F
K462( the carrier of F, the addF of F,(w2 . ys),(w2 . rm)) is Element of the carrier of F
ys is Element of the carrier of (G . (G,m))
rm is V11() real ext-real Element of REAL
rm * ys is Element of the carrier of (G . (G,m))
the Mult of (G . (G,m)) is Relation-like K7(REAL, the carrier of (G . (G,m))) -defined the carrier of (G . (G,m)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (G . (G,m))), the carrier of (G . (G,m))))
K7(REAL, the carrier of (G . (G,m))) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of (G . (G,m))), the carrier of (G . (G,m))) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of (G . (G,m))), the carrier of (G . (G,m)))) is non trivial V49() set
K458( the Mult of (G . (G,m)),rm,ys) is set
w2 . (rm * ys) is Element of the carrier of F
(G,(G,m),(0. (product G))) . (rm * ys) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
Sw1 . ((G,(G,m),(0. (product G))) . (rm * ys)) is Element of the carrier of F
(G,(G,m),(0. (product G))) . ys is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
rm * ((G,(G,m),(0. (product G))) . ys) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
the Mult of (product G) is Relation-like K7(REAL, the carrier of (product G)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (product G)), the carrier of (product G)))
K7(REAL, the carrier of (product G)) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of (product G)), the carrier of (product G)) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of (product G)), the carrier of (product G))) is non trivial V49() set
K458( the Mult of (product G),rm,((G,(G,m),(0. (product G))) . ys)) is set
Sw1 . (rm * ((G,(G,m),(0. (product G))) . ys)) is Element of the carrier of F
Sw1 . ((G,(G,m),(0. (product G))) . ys) is Element of the carrier of F
rm * (Sw1 . ((G,(G,m),(0. (product G))) . ys)) is Element of the carrier of F
the Mult of F is Relation-like K7(REAL, the carrier of F) -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of F), the carrier of F))
K7(REAL, the carrier of F) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of F), the carrier of F) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of F), the carrier of F)) is non trivial V49() set
K458( the Mult of F,rm,(Sw1 . ((G,(G,m),(0. (product G))) . ys))) is set
w2 . ys is Element of the carrier of F
rm * (w2 . ys) is Element of the carrier of F
K458( the Mult of F,rm,(w2 . ys)) is set
rm is V11() real ext-real Element of REAL
rs is Element of the carrier of (G . (G,m))
(G,(G,m),(0. (product G))) . rs is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
Sw1 . ((G,(G,m),(0. (product G))) . rs) is Element of the carrier of F
||.(Sw1 . ((G,(G,m),(0. (product G))) . rs)).|| is V11() real ext-real non negative Element of REAL
the normF of F is non empty Relation-like the carrier of F -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of F,REAL))
K7( the carrier of F,REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of F,REAL)) is non trivial V49() set
the normF of F . (Sw1 . ((G,(G,m),(0. (product G))) . rs)) is V11() real ext-real Element of REAL
||.((G,(G,m),(0. (product G))) . rs).|| is V11() real ext-real non negative Element of REAL
the normF of (product G) is non empty Relation-like the carrier of (product G) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (product G),REAL))
K7( the carrier of (product G),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (product G),REAL)) is non trivial V49() set
the normF of (product G) . ((G,(G,m),(0. (product G))) . rs) is V11() real ext-real Element of REAL
rm * ||.((G,(G,m),(0. (product G))) . rs).|| is V11() real ext-real Element of REAL
||.rs.|| is V11() real ext-real non negative Element of REAL
the normF of (G . (G,m)) is non empty Relation-like the carrier of (G . (G,m)) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (G . (G,m)),REAL))
K7( the carrier of (G . (G,m)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (G . (G,m)),REAL)) is non trivial V49() set
the normF of (G . (G,m)) . rs is V11() real ext-real Element of REAL
rm * ||.rs.|| is V11() real ext-real Element of REAL
ys is non empty Relation-like the carrier of (G . (G,m)) -defined the carrier of F -valued Function-like total quasi_total V153(G . (G,m),F) V154(G . (G,m),F) Element of K6(K7( the carrier of (G . (G,m)), the carrier of F))
ys . rs is Element of the carrier of F
||.(ys . rs).|| is V11() real ext-real non negative Element of REAL
the normF of F . (ys . rs) is V11() real ext-real Element of REAL
i is set
(G,(G,m),X) .: a is Element of K6( the carrier of (product G))
p1 is Element of the carrier of (G . (G,m))
(G,(G,m),X) . p1 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
p1 is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like RestFunc-like Element of K6(K7( the carrier of (product G), the carrier of F))
dom (G,(G,m),X) is non empty Element of K6( the carrier of (G . (G,m)))
dom (f * (G,(G,m),X)) is Element of K6( the carrier of (G . (G,m)))
(f * (G,(G,m),X)) /. y1 is Element of the carrier of F
rs is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F))
tm is Relation-like the carrier of (G . (G,m)) -defined the carrier of F -valued Function-like RestFunc-like Element of K6(K7( the carrier of (G . (G,m)), the carrier of F))
i is Element of the carrier of (G . (G,m))
(f * (G,(G,m),X)) /. i is Element of the carrier of F
((f * (G,(G,m),X)) /. i) - ((f * (G,(G,m),X)) /. y1) is Element of the carrier of F
- ((f * (G,(G,m),X)) /. y1) is Element of the carrier of F
((f * (G,(G,m),X)) /. i) + (- ((f * (G,(G,m),X)) /. y1)) is Element of the carrier of F
K462( the carrier of F, the addF of F,((f * (G,(G,m),X)) /. i),(- ((f * (G,(G,m),X)) /. y1))) is Element of the carrier of F
i - y1 is Element of the carrier of (G . (G,m))
- y1 is Element of the carrier of (G . (G,m))
i + (- y1) is Element of the carrier of (G . (G,m))
K462( the carrier of (G . (G,m)), the addF of (G . (G,m)),i,(- y1)) is Element of the carrier of (G . (G,m))
rs . (i - y1) is Element of the carrier of F
tm /. (i - y1) is Element of the carrier of F
(rs . (i - y1)) + (tm /. (i - y1)) is Element of the carrier of F
K462( the carrier of F, the addF of F,(rs . (i - y1)),(tm /. (i - y1))) is Element of the carrier of F
(G,(G,m),X) . i is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,(G,m),X) . y1 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
Replace ((G,m),y1,) is Relation-like Function-like set
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
X . (G,m) is set
(f * (G,(G,m),X)) . i is set
f . ((G,(G,m),X) . i) is set
f /. ((G,(G,m),X) . i) is Element of the carrier of F
p1 is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like RestFunc-like Element of K6(K7( the carrier of (product G), the carrier of F))
(G,(G,m),(0. (product G))) . (i - y1) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
y0 /. ((G,(G,m),(0. (product G))) . (i - y1)) is Element of the carrier of F
y0 . ((G,(G,m),(0. (product G))) . (i - y1)) is set
tm . (i - y1) is set
(f * (G,(G,m),X)) . y1 is set
f . ((G,(G,m),X) . y1) is set
(f /. ((G,(G,m),X) . i)) - (f /. X) is Element of the carrier of F
- (f /. X) is Element of the carrier of F
(f /. ((G,(G,m),X) . i)) + (- (f /. X)) is Element of the carrier of F
K462( the carrier of F, the addF of F,(f /. ((G,(G,m),X) . i)),(- (f /. X))) is Element of the carrier of F
p1 is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like RestFunc-like Element of K6(K7( the carrier of (product G), the carrier of F))
((G,(G,m),X) . i) - X is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
- X is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((G,(G,m),X) . i) + (- X) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),((G,(G,m),X) . i),(- X)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(diff (f,X)) . (((G,(G,m),X) . i) - X) is Element of the carrier of F
y0 /. (((G,(G,m),X) . i) - X) is Element of the carrier of F
((diff (f,X)) . (((G,(G,m),X) . i) - X)) + (y0 /. (((G,(G,m),X) . i) - X)) is Element of the carrier of F
K462( the carrier of F, the addF of F,((diff (f,X)) . (((G,(G,m),X) . i) - X)),(y0 /. (((G,(G,m),X) . i) - X))) is Element of the carrier of F
Sw1 . ((G,(G,m),(0. (product G))) . (i - y1)) is Element of the carrier of F
(Sw1 . ((G,(G,m),(0. (product G))) . (i - y1))) + (y0 /. (((G,(G,m),X) . i) - X)) is Element of the carrier of F
K462( the carrier of F, the addF of F,(Sw1 . ((G,(G,m),(0. (product G))) . (i - y1))),(y0 /. (((G,(G,m),X) . i) - X))) is Element of the carrier of F
(Sw1 . ((G,(G,m),(0. (product G))) . (i - y1))) + (y0 /. ((G,(G,m),(0. (product G))) . (i - y1))) is Element of the carrier of F
K462( the carrier of F, the addF of F,(Sw1 . ((G,(G,m),(0. (product G))) . (i - y1))),(y0 /. ((G,(G,m),(0. (product G))) . (i - y1)))) is Element of the carrier of F
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len G) is non empty V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= len G ) } is set
F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of F is non empty non trivial set
K7( the carrier of (product G), the carrier of F) is Relation-like set
K6(K7( the carrier of (product G), the carrier of F)) is set
f is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of F))
X is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
i is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
i0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
y0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
(G,y0) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
G . (G,y0) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
(G,F,y0,f,X) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F))
R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,y0)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,y0)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,y0)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,y0)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,y0)),F))
the carrier of (G . (G,y0)) is non empty non trivial set
K266( the carrier of (G . (G,y0)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,y0)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,y0)), the carrier of F
FuncZero ( the carrier of (G . (G,y0)),F) is Relation-like the carrier of (G . (G,y0)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,y0)), the carrier of F)
0. F is V68(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
the carrier of (G . (G,y0)) --> (0. F) is non empty Relation-like the carrier of (G . (G,y0)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,y0)), the carrier of F))
K7( the carrier of (G . (G,y0)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,y0)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,y0)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,y0)), the carrier of F)),(Funcs ( the carrier of (G . (G,y0)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,y0)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,y0)), the carrier of F)),(Funcs ( the carrier of (G . (G,y0)), the carrier of F))),(Funcs ( the carrier of (G . (G,y0)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,y0)), the carrier of F)),(Funcs ( the carrier of (G . (G,y0)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,y0)), the carrier of F)),(Funcs ( the carrier of (G . (G,y0)), the carrier of F))),(Funcs ( the carrier of (G . (G,y0)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,y0)), the carrier of F)),(Funcs ( the carrier of (G . (G,y0)), the carrier of F))),(Funcs ( the carrier of (G . (G,y0)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,y0)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,y0)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,y0)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,y0)), the carrier of F))),(Funcs ( the carrier of (G . (G,y0)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,y0)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,y0)), the carrier of F))),(Funcs ( the carrier of (G . (G,y0)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,y0)), the carrier of F))),(Funcs ( the carrier of (G . (G,y0)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,y0)), the carrier of F)),(FuncZero ( the carrier of (G . (G,y0)),F)),(FuncAdd ( the carrier of (G . (G,y0)),F)),(FuncExtMult ( the carrier of (G . (G,y0)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,y0)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,y0)),F)) is set
K215(K266( the carrier of (G . (G,y0)),F),(LinearOperators ((G . (G,y0)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,y0)),F)
K213(K266( the carrier of (G . (G,y0)),F),(LinearOperators ((G . (G,y0)),F))) is Relation-like K7((LinearOperators ((G . (G,y0)),F)),(LinearOperators ((G . (G,y0)),F))) -defined LinearOperators ((G . (G,y0)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,y0)),F)),(LinearOperators ((G . (G,y0)),F))),(LinearOperators ((G . (G,y0)),F))))
K7((LinearOperators ((G . (G,y0)),F)),(LinearOperators ((G . (G,y0)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,y0)),F)),(LinearOperators ((G . (G,y0)),F))),(LinearOperators ((G . (G,y0)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,y0)),F)),(LinearOperators ((G . (G,y0)),F))),(LinearOperators ((G . (G,y0)),F)))) is set
K214(K266( the carrier of (G . (G,y0)),F),(LinearOperators ((G . (G,y0)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,y0)),F))) -defined LinearOperators ((G . (G,y0)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,y0)),F))),(LinearOperators ((G . (G,y0)),F))))
K7(REAL,(LinearOperators ((G . (G,y0)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,y0)),F))),(LinearOperators ((G . (G,y0)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,y0)),F))),(LinearOperators ((G . (G,y0)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,y0)),F)),K215(K266( the carrier of (G . (G,y0)),F),(LinearOperators ((G . (G,y0)),F))),K213(K266( the carrier of (G . (G,y0)),F),(LinearOperators ((G . (G,y0)),F))),K214(K266( the carrier of (G . (G,y0)),F),(LinearOperators ((G . (G,y0)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,y0)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,y0)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))) is Element of BoundedLinearOperators ((G . (G,y0)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))) -defined BoundedLinearOperators ((G . (G,y0)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))),(BoundedLinearOperators ((G . (G,y0)),F))))
K7((BoundedLinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))),(BoundedLinearOperators ((G . (G,y0)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))),(BoundedLinearOperators ((G . (G,y0)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,y0)),F))) -defined BoundedLinearOperators ((G . (G,y0)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,y0)),F))),(BoundedLinearOperators ((G . (G,y0)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,y0)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,y0)),F))),(BoundedLinearOperators ((G . (G,y0)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,y0)),F))),(BoundedLinearOperators ((G . (G,y0)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,y0)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,y0)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,y0)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,y0)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,y0)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,y0)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))),(BoundedLinearOperatorsNorm ((G . (G,y0)),F)) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F)) is non empty set
(G,(G,y0),X) is non empty Relation-like the carrier of (G . (G,y0)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,y0)), the carrier of (product G)))
K7( the carrier of (G . (G,y0)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,y0)), the carrier of (product G))) is set
f * (G,(G,y0),X) is Relation-like the carrier of (G . (G,y0)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,y0)), the carrier of F))
(G,(G,y0)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,y0)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,y0))))
K7( the carrier of (product G), the carrier of (G . (G,y0))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,y0)))) is set
(G,(G,y0)) . X is Element of the carrier of (G . (G,y0))
diff ((f * (G,(G,y0),X)),((G,(G,y0)) . X)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F))
(G,(G,y0)) . i is Element of the carrier of (G . (G,y0))
(G,F,y0,f,X) . ((G,(G,y0)) . i) is Element of the carrier of F
i0 is Relation-like NAT -defined the carrier of F -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of F
dom i0 is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
y0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
i0 . y0 is set
(G,y0) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
G . (G,y0) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
(G,F,y0,f,X) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F))
R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,y0)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,y0)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,y0)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,y0)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,y0)),F))
the carrier of (G . (G,y0)) is non empty non trivial set
K266( the carrier of (G . (G,y0)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,y0)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,y0)), the carrier of F
FuncZero ( the carrier of (G . (G,y0)),F) is Relation-like the carrier of (G . (G,y0)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,y0)), the carrier of F)
0. F is V68(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
the carrier of (G . (G,y0)) --> (0. F) is non empty Relation-like the carrier of (G . (G,y0)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,y0)), the carrier of F))
K7( the carrier of (G . (G,y0)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,y0)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,y0)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,y0)), the carrier of F)),(Funcs ( the carrier of (G . (G,y0)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,y0)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,y0)), the carrier of F)),(Funcs ( the carrier of (G . (G,y0)), the carrier of F))),(Funcs ( the carrier of (G . (G,y0)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,y0)), the carrier of F)),(Funcs ( the carrier of (G . (G,y0)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,y0)), the carrier of F)),(Funcs ( the carrier of (G . (G,y0)), the carrier of F))),(Funcs ( the carrier of (G . (G,y0)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,y0)), the carrier of F)),(Funcs ( the carrier of (G . (G,y0)), the carrier of F))),(Funcs ( the carrier of (G . (G,y0)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,y0)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,y0)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,y0)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,y0)), the carrier of F))),(Funcs ( the carrier of (G . (G,y0)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,y0)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,y0)), the carrier of F))),(Funcs ( the carrier of (G . (G,y0)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,y0)), the carrier of F))),(Funcs ( the carrier of (G . (G,y0)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,y0)), the carrier of F)),(FuncZero ( the carrier of (G . (G,y0)),F)),(FuncAdd ( the carrier of (G . (G,y0)),F)),(FuncExtMult ( the carrier of (G . (G,y0)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,y0)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,y0)),F)) is set
K215(K266( the carrier of (G . (G,y0)),F),(LinearOperators ((G . (G,y0)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,y0)),F)
K213(K266( the carrier of (G . (G,y0)),F),(LinearOperators ((G . (G,y0)),F))) is Relation-like K7((LinearOperators ((G . (G,y0)),F)),(LinearOperators ((G . (G,y0)),F))) -defined LinearOperators ((G . (G,y0)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,y0)),F)),(LinearOperators ((G . (G,y0)),F))),(LinearOperators ((G . (G,y0)),F))))
K7((LinearOperators ((G . (G,y0)),F)),(LinearOperators ((G . (G,y0)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,y0)),F)),(LinearOperators ((G . (G,y0)),F))),(LinearOperators ((G . (G,y0)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,y0)),F)),(LinearOperators ((G . (G,y0)),F))),(LinearOperators ((G . (G,y0)),F)))) is set
K214(K266( the carrier of (G . (G,y0)),F),(LinearOperators ((G . (G,y0)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,y0)),F))) -defined LinearOperators ((G . (G,y0)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,y0)),F))),(LinearOperators ((G . (G,y0)),F))))
K7(REAL,(LinearOperators ((G . (G,y0)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,y0)),F))),(LinearOperators ((G . (G,y0)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,y0)),F))),(LinearOperators ((G . (G,y0)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,y0)),F)),K215(K266( the carrier of (G . (G,y0)),F),(LinearOperators ((G . (G,y0)),F))),K213(K266( the carrier of (G . (G,y0)),F),(LinearOperators ((G . (G,y0)),F))),K214(K266( the carrier of (G . (G,y0)),F),(LinearOperators ((G . (G,y0)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,y0)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,y0)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))) is Element of BoundedLinearOperators ((G . (G,y0)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))) -defined BoundedLinearOperators ((G . (G,y0)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))),(BoundedLinearOperators ((G . (G,y0)),F))))
K7((BoundedLinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))),(BoundedLinearOperators ((G . (G,y0)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))),(BoundedLinearOperators ((G . (G,y0)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,y0)),F))) -defined BoundedLinearOperators ((G . (G,y0)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,y0)),F))),(BoundedLinearOperators ((G . (G,y0)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,y0)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,y0)),F))),(BoundedLinearOperators ((G . (G,y0)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,y0)),F))),(BoundedLinearOperators ((G . (G,y0)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,y0)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,y0)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,y0)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,y0)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,y0)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,y0)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))),(BoundedLinearOperatorsNorm ((G . (G,y0)),F)) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F)) is non empty set
(G,(G,y0),X) is non empty Relation-like the carrier of (G . (G,y0)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,y0)), the carrier of (product G)))
K7( the carrier of (G . (G,y0)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,y0)), the carrier of (product G))) is set
f * (G,(G,y0),X) is Relation-like the carrier of (G . (G,y0)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,y0)), the carrier of F))
(G,(G,y0)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,y0)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,y0))))
K7( the carrier of (product G), the carrier of (G . (G,y0))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,y0)))) is set
(G,(G,y0)) . X is Element of the carrier of (G . (G,y0))
diff ((f * (G,(G,y0),X)),((G,(G,y0)) . X)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F))
(G,(G,y0)) . i is Element of the carrier of (G . (G,y0))
(G,F,y0,f,X) . ((G,(G,y0)) . i) is Element of the carrier of F
r is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
(G,r) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . (G,r) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
(G,F,r,f,X) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F))
R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,r)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,r)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,r)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,r)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,r)),F))
the carrier of (G . (G,r)) is non empty non trivial set
K266( the carrier of (G . (G,r)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,r)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,r)), the carrier of F
FuncZero ( the carrier of (G . (G,r)),F) is Relation-like the carrier of (G . (G,r)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,r)), the carrier of F)
the carrier of (G . (G,r)) --> (0. F) is non empty Relation-like the carrier of (G . (G,r)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,r)), the carrier of F))
K7( the carrier of (G . (G,r)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,r)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,r)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,r)), the carrier of F)),(Funcs ( the carrier of (G . (G,r)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,r)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,r)), the carrier of F)),(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,r)), the carrier of F)),(Funcs ( the carrier of (G . (G,r)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,r)), the carrier of F)),(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,r)), the carrier of F)),(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,r)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,r)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,r)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,r)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,r)), the carrier of F)),(FuncZero ( the carrier of (G . (G,r)),F)),(FuncAdd ( the carrier of (G . (G,r)),F)),(FuncExtMult ( the carrier of (G . (G,r)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,r)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,r)),F)) is set
K215(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,r)),F)
K213(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))) is Relation-like K7((LinearOperators ((G . (G,r)),F)),(LinearOperators ((G . (G,r)),F))) -defined LinearOperators ((G . (G,r)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,r)),F)),(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F))))
K7((LinearOperators ((G . (G,r)),F)),(LinearOperators ((G . (G,r)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,r)),F)),(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,r)),F)),(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F)))) is set
K214(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,r)),F))) -defined LinearOperators ((G . (G,r)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F))))
K7(REAL,(LinearOperators ((G . (G,r)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,r)),F)),K215(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))),K213(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))),K214(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,r)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,r)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))) is Element of BoundedLinearOperators ((G . (G,r)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))) -defined BoundedLinearOperators ((G . (G,r)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F))))
K7((BoundedLinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,r)),F))) -defined BoundedLinearOperators ((G . (G,r)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,r)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,r)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,r)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,r)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,r)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,r)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,r)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperatorsNorm ((G . (G,r)),F)) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F)) is non empty set
(G,(G,r),X) is non empty Relation-like the carrier of (G . (G,r)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,r)), the carrier of (product G)))
K7( the carrier of (G . (G,r)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,r)), the carrier of (product G))) is set
f * (G,(G,r),X) is Relation-like the carrier of (G . (G,r)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,r)), the carrier of F))
(G,(G,r)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,r)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,r))))
K7( the carrier of (product G), the carrier of (G . (G,r))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,r)))) is set
(G,(G,r)) . X is Element of the carrier of (G . (G,r))
diff ((f * (G,(G,r),X)),((G,(G,r)) . X)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F))
(G,(G,r)) . i is Element of the carrier of (G . (G,r))
(G,F,r,f,X) . ((G,(G,r)) . i) is Element of the carrier of F
Sum i0 is Element of the carrier of F
y0 is Element of the carrier of F
r is Relation-like NAT -defined the carrier of F -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of F
dom r is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
Sum r is Element of the carrier of F
i is non empty Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of F))
i0 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
y0 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
i0 + y0 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
the addF of (product G) is Relation-like K7( the carrier of (product G), the carrier of (product G)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)))
K7( the carrier of (product G), the carrier of (product G)) is Relation-like set
K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)) is Relation-like set
K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G))) is set
K462( the carrier of (product G), the addF of (product G),i0,y0) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
i . (i0 + y0) is Element of the carrier of F
i . i0 is Element of the carrier of F
i . y0 is Element of the carrier of F
(i . i0) + (i . y0) is Element of the carrier of F
the addF of F is Relation-like K7( the carrier of F, the carrier of F) -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of F, the carrier of F), the carrier of F))
K7( the carrier of F, the carrier of F) is Relation-like set
K7(K7( the carrier of F, the carrier of F), the carrier of F) is Relation-like set
K6(K7(K7( the carrier of F, the carrier of F), the carrier of F)) is set
K462( the carrier of F, the addF of F,(i . i0),(i . y0)) is Element of the carrier of F
r is Relation-like NAT -defined the carrier of F -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of F
dom r is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
Sum r is Element of the carrier of F
s is Relation-like NAT -defined the carrier of F -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of F
dom s is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
Sum s is Element of the carrier of F
y1 is Relation-like NAT -defined the carrier of F -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of F
dom y1 is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
Sum y1 is Element of the carrier of F
len r is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
len s is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
len y1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
PP is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
r /. PP is Element of the carrier of F
r . PP is set
(G,PP) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
G . (G,PP) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
(G,F,PP,f,X) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,PP)),F))
R_NormSpace_of_BoundedLinearOperators ((G . (G,PP)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,PP)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,PP)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,PP)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,PP)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,PP)),F))
the carrier of (G . (G,PP)) is non empty non trivial set
K266( the carrier of (G . (G,PP)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,PP)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,PP)), the carrier of F
FuncZero ( the carrier of (G . (G,PP)),F) is Relation-like the carrier of (G . (G,PP)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,PP)), the carrier of F)
0. F is V68(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
the carrier of (G . (G,PP)) --> (0. F) is non empty Relation-like the carrier of (G . (G,PP)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,PP)), the carrier of F))
K7( the carrier of (G . (G,PP)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,PP)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,PP)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,PP)), the carrier of F)),(Funcs ( the carrier of (G . (G,PP)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,PP)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,PP)), the carrier of F)),(Funcs ( the carrier of (G . (G,PP)), the carrier of F))),(Funcs ( the carrier of (G . (G,PP)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,PP)), the carrier of F)),(Funcs ( the carrier of (G . (G,PP)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,PP)), the carrier of F)),(Funcs ( the carrier of (G . (G,PP)), the carrier of F))),(Funcs ( the carrier of (G . (G,PP)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,PP)), the carrier of F)),(Funcs ( the carrier of (G . (G,PP)), the carrier of F))),(Funcs ( the carrier of (G . (G,PP)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,PP)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,PP)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,PP)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,PP)), the carrier of F))),(Funcs ( the carrier of (G . (G,PP)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,PP)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,PP)), the carrier of F))),(Funcs ( the carrier of (G . (G,PP)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,PP)), the carrier of F))),(Funcs ( the carrier of (G . (G,PP)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,PP)), the carrier of F)),(FuncZero ( the carrier of (G . (G,PP)),F)),(FuncAdd ( the carrier of (G . (G,PP)),F)),(FuncExtMult ( the carrier of (G . (G,PP)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,PP)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,PP)),F)) is set
K215(K266( the carrier of (G . (G,PP)),F),(LinearOperators ((G . (G,PP)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,PP)),F)
K213(K266( the carrier of (G . (G,PP)),F),(LinearOperators ((G . (G,PP)),F))) is Relation-like K7((LinearOperators ((G . (G,PP)),F)),(LinearOperators ((G . (G,PP)),F))) -defined LinearOperators ((G . (G,PP)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,PP)),F)),(LinearOperators ((G . (G,PP)),F))),(LinearOperators ((G . (G,PP)),F))))
K7((LinearOperators ((G . (G,PP)),F)),(LinearOperators ((G . (G,PP)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,PP)),F)),(LinearOperators ((G . (G,PP)),F))),(LinearOperators ((G . (G,PP)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,PP)),F)),(LinearOperators ((G . (G,PP)),F))),(LinearOperators ((G . (G,PP)),F)))) is set
K214(K266( the carrier of (G . (G,PP)),F),(LinearOperators ((G . (G,PP)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,PP)),F))) -defined LinearOperators ((G . (G,PP)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,PP)),F))),(LinearOperators ((G . (G,PP)),F))))
K7(REAL,(LinearOperators ((G . (G,PP)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,PP)),F))),(LinearOperators ((G . (G,PP)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,PP)),F))),(LinearOperators ((G . (G,PP)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,PP)),F)),K215(K266( the carrier of (G . (G,PP)),F),(LinearOperators ((G . (G,PP)),F))),K213(K266( the carrier of (G . (G,PP)),F),(LinearOperators ((G . (G,PP)),F))),K214(K266( the carrier of (G . (G,PP)),F),(LinearOperators ((G . (G,PP)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,PP)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,PP)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,PP)),F)),(BoundedLinearOperators ((G . (G,PP)),F))) is Element of BoundedLinearOperators ((G . (G,PP)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,PP)),F)),(BoundedLinearOperators ((G . (G,PP)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,PP)),F)),(BoundedLinearOperators ((G . (G,PP)),F))) -defined BoundedLinearOperators ((G . (G,PP)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,PP)),F)),(BoundedLinearOperators ((G . (G,PP)),F))),(BoundedLinearOperators ((G . (G,PP)),F))))
K7((BoundedLinearOperators ((G . (G,PP)),F)),(BoundedLinearOperators ((G . (G,PP)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,PP)),F)),(BoundedLinearOperators ((G . (G,PP)),F))),(BoundedLinearOperators ((G . (G,PP)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,PP)),F)),(BoundedLinearOperators ((G . (G,PP)),F))),(BoundedLinearOperators ((G . (G,PP)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,PP)),F)),(BoundedLinearOperators ((G . (G,PP)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,PP)),F))) -defined BoundedLinearOperators ((G . (G,PP)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,PP)),F))),(BoundedLinearOperators ((G . (G,PP)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,PP)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,PP)),F))),(BoundedLinearOperators ((G . (G,PP)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,PP)),F))),(BoundedLinearOperators ((G . (G,PP)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,PP)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,PP)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,PP)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,PP)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,PP)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,PP)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,PP)),F)),(BoundedLinearOperators ((G . (G,PP)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,PP)),F)),(BoundedLinearOperators ((G . (G,PP)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,PP)),F)),(BoundedLinearOperators ((G . (G,PP)),F))),(BoundedLinearOperatorsNorm ((G . (G,PP)),F)) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,PP)),F)) is non empty set
(G,(G,PP),X) is non empty Relation-like the carrier of (G . (G,PP)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,PP)), the carrier of (product G)))
K7( the carrier of (G . (G,PP)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,PP)), the carrier of (product G))) is set
f * (G,(G,PP),X) is Relation-like the carrier of (G . (G,PP)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,PP)), the carrier of F))
(G,(G,PP)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,PP)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,PP))))
K7( the carrier of (product G), the carrier of (G . (G,PP))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,PP)))) is set
(G,(G,PP)) . X is Element of the carrier of (G . (G,PP))
diff ((f * (G,(G,PP),X)),((G,(G,PP)) . X)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,PP)),F))
(G,(G,PP)) . i0 is Element of the carrier of (G . (G,PP))
(G,F,PP,f,X) . ((G,(G,PP)) . i0) is Element of the carrier of F
s /. PP is Element of the carrier of F
s . PP is set
(G,(G,PP)) . y0 is Element of the carrier of (G . (G,PP))
(G,F,PP,f,X) . ((G,(G,PP)) . y0) is Element of the carrier of F
y1 . PP is set
(G,(G,PP)) . (i0 + y0) is Element of the carrier of (G . (G,PP))
(G,F,PP,f,X) . ((G,(G,PP)) . (i0 + y0)) is Element of the carrier of F
((G,(G,PP)) . i0) + ((G,(G,PP)) . y0) is Element of the carrier of (G . (G,PP))
the addF of (G . (G,PP)) is Relation-like K7( the carrier of (G . (G,PP)), the carrier of (G . (G,PP))) -defined the carrier of (G . (G,PP)) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . (G,PP)), the carrier of (G . (G,PP))), the carrier of (G . (G,PP))))
K7( the carrier of (G . (G,PP)), the carrier of (G . (G,PP))) is Relation-like set
K7(K7( the carrier of (G . (G,PP)), the carrier of (G . (G,PP))), the carrier of (G . (G,PP))) is Relation-like set
K6(K7(K7( the carrier of (G . (G,PP)), the carrier of (G . (G,PP))), the carrier of (G . (G,PP)))) is set
K462( the carrier of (G . (G,PP)), the addF of (G . (G,PP)),((G,(G,PP)) . i0),((G,(G,PP)) . y0)) is Element of the carrier of (G . (G,PP))
(G,F,PP,f,X) . (((G,(G,PP)) . i0) + ((G,(G,PP)) . y0)) is Element of the carrier of F
(r /. PP) + (s /. PP) is Element of the carrier of F
K462( the carrier of F, the addF of F,(r /. PP),(s /. PP)) is Element of the carrier of F
i0 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
i . i0 is Element of the carrier of F
y0 is V11() real ext-real Element of REAL
y0 * i0 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
the Mult of (product G) is Relation-like K7(REAL, the carrier of (product G)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (product G)), the carrier of (product G)))
K7(REAL, the carrier of (product G)) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of (product G)), the carrier of (product G)) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of (product G)), the carrier of (product G))) is non trivial V49() set
K458( the Mult of (product G),y0,i0) is set
i . (y0 * i0) is Element of the carrier of F
y0 * (i . i0) is Element of the carrier of F
the Mult of F is Relation-like K7(REAL, the carrier of F) -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of F), the carrier of F))
K7(REAL, the carrier of F) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of F), the carrier of F) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of F), the carrier of F)) is non trivial V49() set
K458( the Mult of F,y0,(i . i0)) is set
r is Relation-like NAT -defined the carrier of F -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of F
dom r is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
Sum r is Element of the carrier of F
s is Relation-like NAT -defined the carrier of F -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of F
dom s is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
Sum s is Element of the carrier of F
len r is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
len s is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
y1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
r /. y1 is Element of the carrier of F
r . y1 is set
(G,y1) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
G . (G,y1) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
(G,F,y1,f,X) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y1)),F))
R_NormSpace_of_BoundedLinearOperators ((G . (G,y1)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,y1)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,y1)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,y1)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,y1)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,y1)),F))
the carrier of (G . (G,y1)) is non empty non trivial set
K266( the carrier of (G . (G,y1)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,y1)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,y1)), the carrier of F
FuncZero ( the carrier of (G . (G,y1)),F) is Relation-like the carrier of (G . (G,y1)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,y1)), the carrier of F)
0. F is V68(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
the carrier of (G . (G,y1)) --> (0. F) is non empty Relation-like the carrier of (G . (G,y1)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,y1)), the carrier of F))
K7( the carrier of (G . (G,y1)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,y1)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,y1)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,y1)), the carrier of F)),(Funcs ( the carrier of (G . (G,y1)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,y1)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,y1)), the carrier of F)),(Funcs ( the carrier of (G . (G,y1)), the carrier of F))),(Funcs ( the carrier of (G . (G,y1)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,y1)), the carrier of F)),(Funcs ( the carrier of (G . (G,y1)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,y1)), the carrier of F)),(Funcs ( the carrier of (G . (G,y1)), the carrier of F))),(Funcs ( the carrier of (G . (G,y1)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,y1)), the carrier of F)),(Funcs ( the carrier of (G . (G,y1)), the carrier of F))),(Funcs ( the carrier of (G . (G,y1)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,y1)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,y1)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,y1)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,y1)), the carrier of F))),(Funcs ( the carrier of (G . (G,y1)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,y1)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,y1)), the carrier of F))),(Funcs ( the carrier of (G . (G,y1)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,y1)), the carrier of F))),(Funcs ( the carrier of (G . (G,y1)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,y1)), the carrier of F)),(FuncZero ( the carrier of (G . (G,y1)),F)),(FuncAdd ( the carrier of (G . (G,y1)),F)),(FuncExtMult ( the carrier of (G . (G,y1)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,y1)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,y1)),F)) is set
K215(K266( the carrier of (G . (G,y1)),F),(LinearOperators ((G . (G,y1)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,y1)),F)
K213(K266( the carrier of (G . (G,y1)),F),(LinearOperators ((G . (G,y1)),F))) is Relation-like K7((LinearOperators ((G . (G,y1)),F)),(LinearOperators ((G . (G,y1)),F))) -defined LinearOperators ((G . (G,y1)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,y1)),F)),(LinearOperators ((G . (G,y1)),F))),(LinearOperators ((G . (G,y1)),F))))
K7((LinearOperators ((G . (G,y1)),F)),(LinearOperators ((G . (G,y1)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,y1)),F)),(LinearOperators ((G . (G,y1)),F))),(LinearOperators ((G . (G,y1)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,y1)),F)),(LinearOperators ((G . (G,y1)),F))),(LinearOperators ((G . (G,y1)),F)))) is set
K214(K266( the carrier of (G . (G,y1)),F),(LinearOperators ((G . (G,y1)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,y1)),F))) -defined LinearOperators ((G . (G,y1)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,y1)),F))),(LinearOperators ((G . (G,y1)),F))))
K7(REAL,(LinearOperators ((G . (G,y1)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,y1)),F))),(LinearOperators ((G . (G,y1)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,y1)),F))),(LinearOperators ((G . (G,y1)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,y1)),F)),K215(K266( the carrier of (G . (G,y1)),F),(LinearOperators ((G . (G,y1)),F))),K213(K266( the carrier of (G . (G,y1)),F),(LinearOperators ((G . (G,y1)),F))),K214(K266( the carrier of (G . (G,y1)),F),(LinearOperators ((G . (G,y1)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,y1)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,y1)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,y1)),F)),(BoundedLinearOperators ((G . (G,y1)),F))) is Element of BoundedLinearOperators ((G . (G,y1)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,y1)),F)),(BoundedLinearOperators ((G . (G,y1)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,y1)),F)),(BoundedLinearOperators ((G . (G,y1)),F))) -defined BoundedLinearOperators ((G . (G,y1)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,y1)),F)),(BoundedLinearOperators ((G . (G,y1)),F))),(BoundedLinearOperators ((G . (G,y1)),F))))
K7((BoundedLinearOperators ((G . (G,y1)),F)),(BoundedLinearOperators ((G . (G,y1)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,y1)),F)),(BoundedLinearOperators ((G . (G,y1)),F))),(BoundedLinearOperators ((G . (G,y1)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,y1)),F)),(BoundedLinearOperators ((G . (G,y1)),F))),(BoundedLinearOperators ((G . (G,y1)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,y1)),F)),(BoundedLinearOperators ((G . (G,y1)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,y1)),F))) -defined BoundedLinearOperators ((G . (G,y1)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,y1)),F))),(BoundedLinearOperators ((G . (G,y1)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,y1)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,y1)),F))),(BoundedLinearOperators ((G . (G,y1)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,y1)),F))),(BoundedLinearOperators ((G . (G,y1)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,y1)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,y1)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,y1)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,y1)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,y1)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,y1)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,y1)),F)),(BoundedLinearOperators ((G . (G,y1)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,y1)),F)),(BoundedLinearOperators ((G . (G,y1)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,y1)),F)),(BoundedLinearOperators ((G . (G,y1)),F))),(BoundedLinearOperatorsNorm ((G . (G,y1)),F)) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y1)),F)) is non empty set
(G,(G,y1),X) is non empty Relation-like the carrier of (G . (G,y1)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,y1)), the carrier of (product G)))
K7( the carrier of (G . (G,y1)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,y1)), the carrier of (product G))) is set
f * (G,(G,y1),X) is Relation-like the carrier of (G . (G,y1)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,y1)), the carrier of F))
(G,(G,y1)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,y1)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,y1))))
K7( the carrier of (product G), the carrier of (G . (G,y1))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,y1)))) is set
(G,(G,y1)) . X is Element of the carrier of (G . (G,y1))
diff ((f * (G,(G,y1),X)),((G,(G,y1)) . X)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y1)),F))
(G,(G,y1)) . i0 is Element of the carrier of (G . (G,y1))
(G,F,y1,f,X) . ((G,(G,y1)) . i0) is Element of the carrier of F
s . y1 is set
(G,(G,y1)) . (y0 * i0) is Element of the carrier of (G . (G,y1))
(G,F,y1,f,X) . ((G,(G,y1)) . (y0 * i0)) is Element of the carrier of F
y0 * ((G,(G,y1)) . i0) is Element of the carrier of (G . (G,y1))
the Mult of (G . (G,y1)) is Relation-like K7(REAL, the carrier of (G . (G,y1))) -defined the carrier of (G . (G,y1)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (G . (G,y1))), the carrier of (G . (G,y1))))
K7(REAL, the carrier of (G . (G,y1))) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of (G . (G,y1))), the carrier of (G . (G,y1))) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of (G . (G,y1))), the carrier of (G . (G,y1)))) is non trivial V49() set
K458( the Mult of (G . (G,y1)),y0,((G,(G,y1)) . i0)) is set
(G,F,y1,f,X) . (y0 * ((G,(G,y1)) . i0)) is Element of the carrier of F
y0 * (r /. y1) is Element of the carrier of F
K458( the Mult of F,y0,(r /. y1)) is set
y0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
r is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
(G,r) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
G . (G,r) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,r)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,r)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,r)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,r)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,r)),F))
the carrier of (G . (G,r)) is non empty non trivial set
K266( the carrier of (G . (G,r)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,r)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,r)), the carrier of F
FuncZero ( the carrier of (G . (G,r)),F) is Relation-like the carrier of (G . (G,r)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,r)), the carrier of F)
0. F is V68(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
the carrier of (G . (G,r)) --> (0. F) is non empty Relation-like the carrier of (G . (G,r)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,r)), the carrier of F))
K7( the carrier of (G . (G,r)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,r)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,r)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,r)), the carrier of F)),(Funcs ( the carrier of (G . (G,r)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,r)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,r)), the carrier of F)),(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,r)), the carrier of F)),(Funcs ( the carrier of (G . (G,r)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,r)), the carrier of F)),(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,r)), the carrier of F)),(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,r)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,r)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,r)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,r)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,r)), the carrier of F)),(FuncZero ( the carrier of (G . (G,r)),F)),(FuncAdd ( the carrier of (G . (G,r)),F)),(FuncExtMult ( the carrier of (G . (G,r)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,r)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,r)),F)) is set
K215(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,r)),F)
K213(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))) is Relation-like K7((LinearOperators ((G . (G,r)),F)),(LinearOperators ((G . (G,r)),F))) -defined LinearOperators ((G . (G,r)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,r)),F)),(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F))))
K7((LinearOperators ((G . (G,r)),F)),(LinearOperators ((G . (G,r)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,r)),F)),(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,r)),F)),(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F)))) is set
K214(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,r)),F))) -defined LinearOperators ((G . (G,r)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F))))
K7(REAL,(LinearOperators ((G . (G,r)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,r)),F)),K215(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))),K213(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))),K214(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,r)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,r)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))) is Element of BoundedLinearOperators ((G . (G,r)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))) -defined BoundedLinearOperators ((G . (G,r)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F))))
K7((BoundedLinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,r)),F))) -defined BoundedLinearOperators ((G . (G,r)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,r)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,r)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,r)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,r)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,r)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,r)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,r)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperatorsNorm ((G . (G,r)),F)) #) is strict NORMSTR
(G,F,r,f,X) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F))
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F)) is non empty set
(G,(G,r),X) is non empty Relation-like the carrier of (G . (G,r)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,r)), the carrier of (product G)))
K7( the carrier of (G . (G,r)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,r)), the carrier of (product G))) is set
f * (G,(G,r),X) is Relation-like the carrier of (G . (G,r)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,r)), the carrier of F))
(G,(G,r)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,r)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,r))))
K7( the carrier of (product G), the carrier of (G . (G,r))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,r)))) is set
(G,(G,r)) . X is Element of the carrier of (G . (G,r))
diff ((f * (G,(G,r),X)),((G,(G,r)) . X)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F))
||.(G,F,r,f,X).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F)) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F)) -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 ((G . (G,r)),F)),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F)),REAL)) is non trivial V49() set
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F)) . (G,F,r,f,X) is V11() real ext-real Element of REAL
s is V11() real ext-real Element of REAL
y1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
(G,y1) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . (G,y1) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
R_NormSpace_of_BoundedLinearOperators ((G . (G,y1)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,y1)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,y1)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,y1)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,y1)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,y1)),F))
the carrier of (G . (G,y1)) is non empty non trivial set
K266( the carrier of (G . (G,y1)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,y1)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,y1)), the carrier of F
FuncZero ( the carrier of (G . (G,y1)),F) is Relation-like the carrier of (G . (G,y1)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,y1)), the carrier of F)
the carrier of (G . (G,y1)) --> (0. F) is non empty Relation-like the carrier of (G . (G,y1)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,y1)), the carrier of F))
K7( the carrier of (G . (G,y1)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,y1)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,y1)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,y1)), the carrier of F)),(Funcs ( the carrier of (G . (G,y1)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,y1)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,y1)), the carrier of F)),(Funcs ( the carrier of (G . (G,y1)), the carrier of F))),(Funcs ( the carrier of (G . (G,y1)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,y1)), the carrier of F)),(Funcs ( the carrier of (G . (G,y1)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,y1)), the carrier of F)),(Funcs ( the carrier of (G . (G,y1)), the carrier of F))),(Funcs ( the carrier of (G . (G,y1)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,y1)), the carrier of F)),(Funcs ( the carrier of (G . (G,y1)), the carrier of F))),(Funcs ( the carrier of (G . (G,y1)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,y1)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,y1)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,y1)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,y1)), the carrier of F))),(Funcs ( the carrier of (G . (G,y1)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,y1)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,y1)), the carrier of F))),(Funcs ( the carrier of (G . (G,y1)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,y1)), the carrier of F))),(Funcs ( the carrier of (G . (G,y1)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,y1)), the carrier of F)),(FuncZero ( the carrier of (G . (G,y1)),F)),(FuncAdd ( the carrier of (G . (G,y1)),F)),(FuncExtMult ( the carrier of (G . (G,y1)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,y1)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,y1)),F)) is set
K215(K266( the carrier of (G . (G,y1)),F),(LinearOperators ((G . (G,y1)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,y1)),F)
K213(K266( the carrier of (G . (G,y1)),F),(LinearOperators ((G . (G,y1)),F))) is Relation-like K7((LinearOperators ((G . (G,y1)),F)),(LinearOperators ((G . (G,y1)),F))) -defined LinearOperators ((G . (G,y1)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,y1)),F)),(LinearOperators ((G . (G,y1)),F))),(LinearOperators ((G . (G,y1)),F))))
K7((LinearOperators ((G . (G,y1)),F)),(LinearOperators ((G . (G,y1)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,y1)),F)),(LinearOperators ((G . (G,y1)),F))),(LinearOperators ((G . (G,y1)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,y1)),F)),(LinearOperators ((G . (G,y1)),F))),(LinearOperators ((G . (G,y1)),F)))) is set
K214(K266( the carrier of (G . (G,y1)),F),(LinearOperators ((G . (G,y1)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,y1)),F))) -defined LinearOperators ((G . (G,y1)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,y1)),F))),(LinearOperators ((G . (G,y1)),F))))
K7(REAL,(LinearOperators ((G . (G,y1)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,y1)),F))),(LinearOperators ((G . (G,y1)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,y1)),F))),(LinearOperators ((G . (G,y1)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,y1)),F)),K215(K266( the carrier of (G . (G,y1)),F),(LinearOperators ((G . (G,y1)),F))),K213(K266( the carrier of (G . (G,y1)),F),(LinearOperators ((G . (G,y1)),F))),K214(K266( the carrier of (G . (G,y1)),F),(LinearOperators ((G . (G,y1)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,y1)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,y1)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,y1)),F)),(BoundedLinearOperators ((G . (G,y1)),F))) is Element of BoundedLinearOperators ((G . (G,y1)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,y1)),F)),(BoundedLinearOperators ((G . (G,y1)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,y1)),F)),(BoundedLinearOperators ((G . (G,y1)),F))) -defined BoundedLinearOperators ((G . (G,y1)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,y1)),F)),(BoundedLinearOperators ((G . (G,y1)),F))),(BoundedLinearOperators ((G . (G,y1)),F))))
K7((BoundedLinearOperators ((G . (G,y1)),F)),(BoundedLinearOperators ((G . (G,y1)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,y1)),F)),(BoundedLinearOperators ((G . (G,y1)),F))),(BoundedLinearOperators ((G . (G,y1)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,y1)),F)),(BoundedLinearOperators ((G . (G,y1)),F))),(BoundedLinearOperators ((G . (G,y1)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,y1)),F)),(BoundedLinearOperators ((G . (G,y1)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,y1)),F))) -defined BoundedLinearOperators ((G . (G,y1)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,y1)),F))),(BoundedLinearOperators ((G . (G,y1)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,y1)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,y1)),F))),(BoundedLinearOperators ((G . (G,y1)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,y1)),F))),(BoundedLinearOperators ((G . (G,y1)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,y1)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,y1)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,y1)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,y1)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,y1)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,y1)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,y1)),F)),(BoundedLinearOperators ((G . (G,y1)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,y1)),F)),(BoundedLinearOperators ((G . (G,y1)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,y1)),F)),(BoundedLinearOperators ((G . (G,y1)),F))),(BoundedLinearOperatorsNorm ((G . (G,y1)),F)) #) is strict NORMSTR
(G,F,y1,f,X) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y1)),F))
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y1)),F)) is non empty set
(G,(G,y1),X) is non empty Relation-like the carrier of (G . (G,y1)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,y1)), the carrier of (product G)))
K7( the carrier of (G . (G,y1)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,y1)), the carrier of (product G))) is set
f * (G,(G,y1),X) is Relation-like the carrier of (G . (G,y1)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,y1)), the carrier of F))
(G,(G,y1)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,y1)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,y1))))
K7( the carrier of (product G), the carrier of (G . (G,y1))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,y1)))) is set
(G,(G,y1)) . X is Element of the carrier of (G . (G,y1))
diff ((f * (G,(G,y1),X)),((G,(G,y1)) . X)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y1)),F))
||.(G,F,y1,f,X).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y1)),F)) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y1)),F)) -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 ((G . (G,y1)),F)),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y1)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y1)),F)),REAL)) is non trivial V49() set
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y1)),F)) . (G,F,y1,f,X) is V11() real ext-real Element of REAL
y0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
dom y0 is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
r is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
y0 . r is V11() real ext-real Element of REAL
(G,r) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
G . (G,r) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,r)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,r)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,r)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,r)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,r)),F))
the carrier of (G . (G,r)) is non empty non trivial set
K266( the carrier of (G . (G,r)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,r)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,r)), the carrier of F
FuncZero ( the carrier of (G . (G,r)),F) is Relation-like the carrier of (G . (G,r)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,r)), the carrier of F)
0. F is V68(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
the carrier of (G . (G,r)) --> (0. F) is non empty Relation-like the carrier of (G . (G,r)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,r)), the carrier of F))
K7( the carrier of (G . (G,r)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,r)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,r)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,r)), the carrier of F)),(Funcs ( the carrier of (G . (G,r)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,r)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,r)), the carrier of F)),(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,r)), the carrier of F)),(Funcs ( the carrier of (G . (G,r)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,r)), the carrier of F)),(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,r)), the carrier of F)),(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,r)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,r)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,r)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,r)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,r)), the carrier of F)),(FuncZero ( the carrier of (G . (G,r)),F)),(FuncAdd ( the carrier of (G . (G,r)),F)),(FuncExtMult ( the carrier of (G . (G,r)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,r)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,r)),F)) is set
K215(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,r)),F)
K213(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))) is Relation-like K7((LinearOperators ((G . (G,r)),F)),(LinearOperators ((G . (G,r)),F))) -defined LinearOperators ((G . (G,r)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,r)),F)),(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F))))
K7((LinearOperators ((G . (G,r)),F)),(LinearOperators ((G . (G,r)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,r)),F)),(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,r)),F)),(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F)))) is set
K214(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,r)),F))) -defined LinearOperators ((G . (G,r)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F))))
K7(REAL,(LinearOperators ((G . (G,r)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,r)),F)),K215(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))),K213(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))),K214(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,r)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,r)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))) is Element of BoundedLinearOperators ((G . (G,r)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))) -defined BoundedLinearOperators ((G . (G,r)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F))))
K7((BoundedLinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,r)),F))) -defined BoundedLinearOperators ((G . (G,r)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,r)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,r)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,r)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,r)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,r)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,r)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,r)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperatorsNorm ((G . (G,r)),F)) #) is strict NORMSTR
(G,F,r,f,X) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F))
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F)) is non empty set
(G,(G,r),X) is non empty Relation-like the carrier of (G . (G,r)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,r)), the carrier of (product G)))
K7( the carrier of (G . (G,r)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,r)), the carrier of (product G))) is set
f * (G,(G,r),X) is Relation-like the carrier of (G . (G,r)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,r)), the carrier of F))
(G,(G,r)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,r)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,r))))
K7( the carrier of (product G), the carrier of (G . (G,r))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,r)))) is set
(G,(G,r)) . X is Element of the carrier of (G . (G,r))
diff ((f * (G,(G,r),X)),((G,(G,r)) . X)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F))
||.(G,F,r,f,X).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F)) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F)) -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 ((G . (G,r)),F)),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F)),REAL)) is non trivial V49() set
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F)) . (G,F,r,f,X) is V11() real ext-real Element of REAL
s is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
(G,s) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . (G,s) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
R_NormSpace_of_BoundedLinearOperators ((G . (G,s)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,s)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,s)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,s)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,s)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,s)),F))
the carrier of (G . (G,s)) is non empty non trivial set
K266( the carrier of (G . (G,s)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,s)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,s)), the carrier of F
FuncZero ( the carrier of (G . (G,s)),F) is Relation-like the carrier of (G . (G,s)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,s)), the carrier of F)
the carrier of (G . (G,s)) --> (0. F) is non empty Relation-like the carrier of (G . (G,s)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,s)), the carrier of F))
K7( the carrier of (G . (G,s)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,s)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,s)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,s)), the carrier of F)),(Funcs ( the carrier of (G . (G,s)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,s)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,s)), the carrier of F)),(Funcs ( the carrier of (G . (G,s)), the carrier of F))),(Funcs ( the carrier of (G . (G,s)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,s)), the carrier of F)),(Funcs ( the carrier of (G . (G,s)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,s)), the carrier of F)),(Funcs ( the carrier of (G . (G,s)), the carrier of F))),(Funcs ( the carrier of (G . (G,s)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,s)), the carrier of F)),(Funcs ( the carrier of (G . (G,s)), the carrier of F))),(Funcs ( the carrier of (G . (G,s)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,s)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,s)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,s)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,s)), the carrier of F))),(Funcs ( the carrier of (G . (G,s)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,s)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,s)), the carrier of F))),(Funcs ( the carrier of (G . (G,s)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,s)), the carrier of F))),(Funcs ( the carrier of (G . (G,s)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,s)), the carrier of F)),(FuncZero ( the carrier of (G . (G,s)),F)),(FuncAdd ( the carrier of (G . (G,s)),F)),(FuncExtMult ( the carrier of (G . (G,s)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,s)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,s)),F)) is set
K215(K266( the carrier of (G . (G,s)),F),(LinearOperators ((G . (G,s)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,s)),F)
K213(K266( the carrier of (G . (G,s)),F),(LinearOperators ((G . (G,s)),F))) is Relation-like K7((LinearOperators ((G . (G,s)),F)),(LinearOperators ((G . (G,s)),F))) -defined LinearOperators ((G . (G,s)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,s)),F)),(LinearOperators ((G . (G,s)),F))),(LinearOperators ((G . (G,s)),F))))
K7((LinearOperators ((G . (G,s)),F)),(LinearOperators ((G . (G,s)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,s)),F)),(LinearOperators ((G . (G,s)),F))),(LinearOperators ((G . (G,s)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,s)),F)),(LinearOperators ((G . (G,s)),F))),(LinearOperators ((G . (G,s)),F)))) is set
K214(K266( the carrier of (G . (G,s)),F),(LinearOperators ((G . (G,s)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,s)),F))) -defined LinearOperators ((G . (G,s)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,s)),F))),(LinearOperators ((G . (G,s)),F))))
K7(REAL,(LinearOperators ((G . (G,s)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,s)),F))),(LinearOperators ((G . (G,s)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,s)),F))),(LinearOperators ((G . (G,s)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,s)),F)),K215(K266( the carrier of (G . (G,s)),F),(LinearOperators ((G . (G,s)),F))),K213(K266( the carrier of (G . (G,s)),F),(LinearOperators ((G . (G,s)),F))),K214(K266( the carrier of (G . (G,s)),F),(LinearOperators ((G . (G,s)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,s)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,s)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,s)),F)),(BoundedLinearOperators ((G . (G,s)),F))) is Element of BoundedLinearOperators ((G . (G,s)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,s)),F)),(BoundedLinearOperators ((G . (G,s)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,s)),F)),(BoundedLinearOperators ((G . (G,s)),F))) -defined BoundedLinearOperators ((G . (G,s)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,s)),F)),(BoundedLinearOperators ((G . (G,s)),F))),(BoundedLinearOperators ((G . (G,s)),F))))
K7((BoundedLinearOperators ((G . (G,s)),F)),(BoundedLinearOperators ((G . (G,s)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,s)),F)),(BoundedLinearOperators ((G . (G,s)),F))),(BoundedLinearOperators ((G . (G,s)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,s)),F)),(BoundedLinearOperators ((G . (G,s)),F))),(BoundedLinearOperators ((G . (G,s)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,s)),F)),(BoundedLinearOperators ((G . (G,s)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,s)),F))) -defined BoundedLinearOperators ((G . (G,s)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,s)),F))),(BoundedLinearOperators ((G . (G,s)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,s)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,s)),F))),(BoundedLinearOperators ((G . (G,s)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,s)),F))),(BoundedLinearOperators ((G . (G,s)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,s)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,s)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,s)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,s)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,s)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,s)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,s)),F)),(BoundedLinearOperators ((G . (G,s)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,s)),F)),(BoundedLinearOperators ((G . (G,s)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,s)),F)),(BoundedLinearOperators ((G . (G,s)),F))),(BoundedLinearOperatorsNorm ((G . (G,s)),F)) #) is strict NORMSTR
(G,F,s,f,X) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,s)),F))
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,s)),F)) is non empty set
(G,(G,s),X) is non empty Relation-like the carrier of (G . (G,s)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,s)), the carrier of (product G)))
K7( the carrier of (G . (G,s)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,s)), the carrier of (product G))) is set
f * (G,(G,s),X) is Relation-like the carrier of (G . (G,s)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,s)), the carrier of F))
(G,(G,s)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,s)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,s))))
K7( the carrier of (product G), the carrier of (G . (G,s))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,s)))) is set
(G,(G,s)) . X is Element of the carrier of (G . (G,s))
diff ((f * (G,(G,s),X)),((G,(G,s)) . X)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,s)),F))
||.(G,F,s,f,X).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,s)),F)) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,s)),F)) -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 ((G . (G,s)),F)),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,s)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,s)),F)),REAL)) is non trivial V49() set
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,s)),F)) . (G,F,s,f,X) is V11() real ext-real Element of REAL
r is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
y0 . r is V11() real ext-real Element of REAL
(G,r) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . (G,r) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,r)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,r)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,r)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,r)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,r)),F))
the carrier of (G . (G,r)) is non empty non trivial set
K266( the carrier of (G . (G,r)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,r)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,r)), the carrier of F
FuncZero ( the carrier of (G . (G,r)),F) is Relation-like the carrier of (G . (G,r)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,r)), the carrier of F)
the carrier of (G . (G,r)) --> (0. F) is non empty Relation-like the carrier of (G . (G,r)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,r)), the carrier of F))
K7( the carrier of (G . (G,r)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,r)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,r)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,r)), the carrier of F)),(Funcs ( the carrier of (G . (G,r)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,r)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,r)), the carrier of F)),(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,r)), the carrier of F)),(Funcs ( the carrier of (G . (G,r)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,r)), the carrier of F)),(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,r)), the carrier of F)),(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,r)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,r)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,r)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,r)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,r)), the carrier of F)),(FuncZero ( the carrier of (G . (G,r)),F)),(FuncAdd ( the carrier of (G . (G,r)),F)),(FuncExtMult ( the carrier of (G . (G,r)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,r)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,r)),F)) is set
K215(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,r)),F)
K213(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))) is Relation-like K7((LinearOperators ((G . (G,r)),F)),(LinearOperators ((G . (G,r)),F))) -defined LinearOperators ((G . (G,r)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,r)),F)),(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F))))
K7((LinearOperators ((G . (G,r)),F)),(LinearOperators ((G . (G,r)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,r)),F)),(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,r)),F)),(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F)))) is set
K214(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,r)),F))) -defined LinearOperators ((G . (G,r)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F))))
K7(REAL,(LinearOperators ((G . (G,r)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,r)),F)),K215(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))),K213(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))),K214(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,r)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,r)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))) is Element of BoundedLinearOperators ((G . (G,r)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))) -defined BoundedLinearOperators ((G . (G,r)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F))))
K7((BoundedLinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,r)),F))) -defined BoundedLinearOperators ((G . (G,r)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,r)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,r)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,r)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,r)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,r)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,r)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,r)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperatorsNorm ((G . (G,r)),F)) #) is strict NORMSTR
(G,F,r,f,X) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F))
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F)) is non empty set
(G,(G,r),X) is non empty Relation-like the carrier of (G . (G,r)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,r)), the carrier of (product G)))
K7( the carrier of (G . (G,r)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,r)), the carrier of (product G))) is set
f * (G,(G,r),X) is Relation-like the carrier of (G . (G,r)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,r)), the carrier of F))
(G,(G,r)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,r)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,r))))
K7( the carrier of (product G), the carrier of (G . (G,r))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,r)))) is set
(G,(G,r)) . X is Element of the carrier of (G . (G,r))
diff ((f * (G,(G,r),X)),((G,(G,r)) . X)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F))
||.(G,F,r,f,X).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F)) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F)) -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 ((G . (G,r)),F)),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F)),REAL)) is non trivial V49() set
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F)) . (G,F,r,f,X) is V11() real ext-real Element of REAL
Sum y0 is V11() real ext-real Element of REAL
i0 is non empty Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like total quasi_total V153( product G,F) V154( product G,F) Element of K6(K7( the carrier of (product G), the carrier of F))
s is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
i0 . s is Element of the carrier of F
||.(i0 . s).|| is V11() real ext-real non negative Element of REAL
the normF of F is non empty Relation-like the carrier of F -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of F,REAL))
K7( the carrier of F,REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of F,REAL)) is non trivial V49() set
the normF of F . (i0 . s) is V11() real ext-real Element of REAL
||.s.|| is V11() real ext-real non negative Element of REAL
the normF of (product G) is non empty Relation-like the carrier of (product G) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (product G),REAL))
K7( the carrier of (product G),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (product G),REAL)) is non trivial V49() set
the normF of (product G) . s is V11() real ext-real Element of REAL
(Sum y0) * ||.s.|| is V11() real ext-real Element of REAL
y1 is Relation-like NAT -defined the carrier of F -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of F
dom y1 is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
Sum y1 is Element of the carrier of F
PP is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
DD is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
(G,DD) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . (G,DD) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,DD)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,DD)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,DD)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,DD)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,DD)),F))
the carrier of (G . (G,DD)) is non empty non trivial set
K266( the carrier of (G . (G,DD)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,DD)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,DD)), the carrier of F
FuncZero ( the carrier of (G . (G,DD)),F) is Relation-like the carrier of (G . (G,DD)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,DD)), the carrier of F)
the carrier of (G . (G,DD)) --> (0. F) is non empty Relation-like the carrier of (G . (G,DD)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,DD)), the carrier of F))
K7( the carrier of (G . (G,DD)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,DD)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,DD)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,DD)), the carrier of F)),(Funcs ( the carrier of (G . (G,DD)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,DD)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,DD)), the carrier of F)),(Funcs ( the carrier of (G . (G,DD)), the carrier of F))),(Funcs ( the carrier of (G . (G,DD)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,DD)), the carrier of F)),(Funcs ( the carrier of (G . (G,DD)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,DD)), the carrier of F)),(Funcs ( the carrier of (G . (G,DD)), the carrier of F))),(Funcs ( the carrier of (G . (G,DD)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,DD)), the carrier of F)),(Funcs ( the carrier of (G . (G,DD)), the carrier of F))),(Funcs ( the carrier of (G . (G,DD)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,DD)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,DD)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,DD)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,DD)), the carrier of F))),(Funcs ( the carrier of (G . (G,DD)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,DD)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,DD)), the carrier of F))),(Funcs ( the carrier of (G . (G,DD)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,DD)), the carrier of F))),(Funcs ( the carrier of (G . (G,DD)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,DD)), the carrier of F)),(FuncZero ( the carrier of (G . (G,DD)),F)),(FuncAdd ( the carrier of (G . (G,DD)),F)),(FuncExtMult ( the carrier of (G . (G,DD)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,DD)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,DD)),F)) is set
K215(K266( the carrier of (G . (G,DD)),F),(LinearOperators ((G . (G,DD)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,DD)),F)
K213(K266( the carrier of (G . (G,DD)),F),(LinearOperators ((G . (G,DD)),F))) is Relation-like K7((LinearOperators ((G . (G,DD)),F)),(LinearOperators ((G . (G,DD)),F))) -defined LinearOperators ((G . (G,DD)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,DD)),F)),(LinearOperators ((G . (G,DD)),F))),(LinearOperators ((G . (G,DD)),F))))
K7((LinearOperators ((G . (G,DD)),F)),(LinearOperators ((G . (G,DD)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,DD)),F)),(LinearOperators ((G . (G,DD)),F))),(LinearOperators ((G . (G,DD)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,DD)),F)),(LinearOperators ((G . (G,DD)),F))),(LinearOperators ((G . (G,DD)),F)))) is set
K214(K266( the carrier of (G . (G,DD)),F),(LinearOperators ((G . (G,DD)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,DD)),F))) -defined LinearOperators ((G . (G,DD)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,DD)),F))),(LinearOperators ((G . (G,DD)),F))))
K7(REAL,(LinearOperators ((G . (G,DD)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,DD)),F))),(LinearOperators ((G . (G,DD)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,DD)),F))),(LinearOperators ((G . (G,DD)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,DD)),F)),K215(K266( the carrier of (G . (G,DD)),F),(LinearOperators ((G . (G,DD)),F))),K213(K266( the carrier of (G . (G,DD)),F),(LinearOperators ((G . (G,DD)),F))),K214(K266( the carrier of (G . (G,DD)),F),(LinearOperators ((G . (G,DD)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,DD)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,DD)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))) is Element of BoundedLinearOperators ((G . (G,DD)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))) -defined BoundedLinearOperators ((G . (G,DD)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))),(BoundedLinearOperators ((G . (G,DD)),F))))
K7((BoundedLinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))),(BoundedLinearOperators ((G . (G,DD)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))),(BoundedLinearOperators ((G . (G,DD)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,DD)),F))) -defined BoundedLinearOperators ((G . (G,DD)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,DD)),F))),(BoundedLinearOperators ((G . (G,DD)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,DD)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,DD)),F))),(BoundedLinearOperators ((G . (G,DD)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,DD)),F))),(BoundedLinearOperators ((G . (G,DD)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,DD)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,DD)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,DD)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,DD)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,DD)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,DD)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))),(BoundedLinearOperatorsNorm ((G . (G,DD)),F)) #) is strict NORMSTR
(G,F,DD,f,X) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F))
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)) is non empty set
(G,(G,DD),X) is non empty Relation-like the carrier of (G . (G,DD)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,DD)), the carrier of (product G)))
K7( the carrier of (G . (G,DD)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,DD)), the carrier of (product G))) is set
f * (G,(G,DD),X) is Relation-like the carrier of (G . (G,DD)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,DD)), the carrier of F))
(G,(G,DD)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,DD)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,DD))))
K7( the carrier of (product G), the carrier of (G . (G,DD))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,DD)))) is set
(G,(G,DD)) . X is Element of the carrier of (G . (G,DD))
diff ((f * (G,(G,DD),X)),((G,(G,DD)) . X)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F))
||.(G,F,DD,f,X).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)) -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 ((G . (G,DD)),F)),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)),REAL)) is non trivial V49() set
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)) . (G,F,DD,f,X) is V11() real ext-real Element of REAL
||.(G,F,DD,f,X).|| * ||.s.|| is V11() real ext-real non negative Element of REAL
a is V11() real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
(G,t) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . (G,t) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
R_NormSpace_of_BoundedLinearOperators ((G . (G,t)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,t)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,t)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,t)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,t)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,t)),F))
the carrier of (G . (G,t)) is non empty non trivial set
K266( the carrier of (G . (G,t)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,t)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,t)), the carrier of F
FuncZero ( the carrier of (G . (G,t)),F) is Relation-like the carrier of (G . (G,t)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,t)), the carrier of F)
the carrier of (G . (G,t)) --> (0. F) is non empty Relation-like the carrier of (G . (G,t)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,t)), the carrier of F))
K7( the carrier of (G . (G,t)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,t)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,t)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,t)), the carrier of F)),(Funcs ( the carrier of (G . (G,t)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,t)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,t)), the carrier of F)),(Funcs ( the carrier of (G . (G,t)), the carrier of F))),(Funcs ( the carrier of (G . (G,t)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,t)), the carrier of F)),(Funcs ( the carrier of (G . (G,t)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,t)), the carrier of F)),(Funcs ( the carrier of (G . (G,t)), the carrier of F))),(Funcs ( the carrier of (G . (G,t)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,t)), the carrier of F)),(Funcs ( the carrier of (G . (G,t)), the carrier of F))),(Funcs ( the carrier of (G . (G,t)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,t)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,t)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,t)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,t)), the carrier of F))),(Funcs ( the carrier of (G . (G,t)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,t)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,t)), the carrier of F))),(Funcs ( the carrier of (G . (G,t)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,t)), the carrier of F))),(Funcs ( the carrier of (G . (G,t)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,t)), the carrier of F)),(FuncZero ( the carrier of (G . (G,t)),F)),(FuncAdd ( the carrier of (G . (G,t)),F)),(FuncExtMult ( the carrier of (G . (G,t)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,t)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,t)),F)) is set
K215(K266( the carrier of (G . (G,t)),F),(LinearOperators ((G . (G,t)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,t)),F)
K213(K266( the carrier of (G . (G,t)),F),(LinearOperators ((G . (G,t)),F))) is Relation-like K7((LinearOperators ((G . (G,t)),F)),(LinearOperators ((G . (G,t)),F))) -defined LinearOperators ((G . (G,t)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,t)),F)),(LinearOperators ((G . (G,t)),F))),(LinearOperators ((G . (G,t)),F))))
K7((LinearOperators ((G . (G,t)),F)),(LinearOperators ((G . (G,t)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,t)),F)),(LinearOperators ((G . (G,t)),F))),(LinearOperators ((G . (G,t)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,t)),F)),(LinearOperators ((G . (G,t)),F))),(LinearOperators ((G . (G,t)),F)))) is set
K214(K266( the carrier of (G . (G,t)),F),(LinearOperators ((G . (G,t)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,t)),F))) -defined LinearOperators ((G . (G,t)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,t)),F))),(LinearOperators ((G . (G,t)),F))))
K7(REAL,(LinearOperators ((G . (G,t)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,t)),F))),(LinearOperators ((G . (G,t)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,t)),F))),(LinearOperators ((G . (G,t)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,t)),F)),K215(K266( the carrier of (G . (G,t)),F),(LinearOperators ((G . (G,t)),F))),K213(K266( the carrier of (G . (G,t)),F),(LinearOperators ((G . (G,t)),F))),K214(K266( the carrier of (G . (G,t)),F),(LinearOperators ((G . (G,t)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,t)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,t)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,t)),F)),(BoundedLinearOperators ((G . (G,t)),F))) is Element of BoundedLinearOperators ((G . (G,t)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,t)),F)),(BoundedLinearOperators ((G . (G,t)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,t)),F)),(BoundedLinearOperators ((G . (G,t)),F))) -defined BoundedLinearOperators ((G . (G,t)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,t)),F)),(BoundedLinearOperators ((G . (G,t)),F))),(BoundedLinearOperators ((G . (G,t)),F))))
K7((BoundedLinearOperators ((G . (G,t)),F)),(BoundedLinearOperators ((G . (G,t)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,t)),F)),(BoundedLinearOperators ((G . (G,t)),F))),(BoundedLinearOperators ((G . (G,t)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,t)),F)),(BoundedLinearOperators ((G . (G,t)),F))),(BoundedLinearOperators ((G . (G,t)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,t)),F)),(BoundedLinearOperators ((G . (G,t)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,t)),F))) -defined BoundedLinearOperators ((G . (G,t)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,t)),F))),(BoundedLinearOperators ((G . (G,t)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,t)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,t)),F))),(BoundedLinearOperators ((G . (G,t)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,t)),F))),(BoundedLinearOperators ((G . (G,t)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,t)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,t)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,t)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,t)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,t)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,t)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,t)),F)),(BoundedLinearOperators ((G . (G,t)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,t)),F)),(BoundedLinearOperators ((G . (G,t)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,t)),F)),(BoundedLinearOperators ((G . (G,t)),F))),(BoundedLinearOperatorsNorm ((G . (G,t)),F)) #) is strict NORMSTR
(G,F,t,f,X) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,t)),F))
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,t)),F)) is non empty set
(G,(G,t),X) is non empty Relation-like the carrier of (G . (G,t)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,t)), the carrier of (product G)))
K7( the carrier of (G . (G,t)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,t)), the carrier of (product G))) is set
f * (G,(G,t),X) is Relation-like the carrier of (G . (G,t)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,t)), the carrier of F))
(G,(G,t)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,t)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,t))))
K7( the carrier of (product G), the carrier of (G . (G,t))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,t)))) is set
(G,(G,t)) . X is Element of the carrier of (G . (G,t))
diff ((f * (G,(G,t),X)),((G,(G,t)) . X)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,t)),F))
||.(G,F,t,f,X).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,t)),F)) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,t)),F)) -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 ((G . (G,t)),F)),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,t)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,t)),F)),REAL)) is non trivial V49() set
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,t)),F)) . (G,F,t,f,X) is V11() real ext-real Element of REAL
||.(G,F,t,f,X).|| * ||.s.|| is V11() real ext-real non negative Element of REAL
PP is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
dom PP is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
DD is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
PP . DD is V11() real ext-real Element of REAL
(G,DD) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . (G,DD) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,DD)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,DD)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,DD)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,DD)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,DD)),F))
the carrier of (G . (G,DD)) is non empty non trivial set
K266( the carrier of (G . (G,DD)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,DD)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,DD)), the carrier of F
FuncZero ( the carrier of (G . (G,DD)),F) is Relation-like the carrier of (G . (G,DD)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,DD)), the carrier of F)
the carrier of (G . (G,DD)) --> (0. F) is non empty Relation-like the carrier of (G . (G,DD)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,DD)), the carrier of F))
K7( the carrier of (G . (G,DD)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,DD)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,DD)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,DD)), the carrier of F)),(Funcs ( the carrier of (G . (G,DD)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,DD)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,DD)), the carrier of F)),(Funcs ( the carrier of (G . (G,DD)), the carrier of F))),(Funcs ( the carrier of (G . (G,DD)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,DD)), the carrier of F)),(Funcs ( the carrier of (G . (G,DD)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,DD)), the carrier of F)),(Funcs ( the carrier of (G . (G,DD)), the carrier of F))),(Funcs ( the carrier of (G . (G,DD)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,DD)), the carrier of F)),(Funcs ( the carrier of (G . (G,DD)), the carrier of F))),(Funcs ( the carrier of (G . (G,DD)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,DD)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,DD)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,DD)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,DD)), the carrier of F))),(Funcs ( the carrier of (G . (G,DD)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,DD)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,DD)), the carrier of F))),(Funcs ( the carrier of (G . (G,DD)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,DD)), the carrier of F))),(Funcs ( the carrier of (G . (G,DD)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,DD)), the carrier of F)),(FuncZero ( the carrier of (G . (G,DD)),F)),(FuncAdd ( the carrier of (G . (G,DD)),F)),(FuncExtMult ( the carrier of (G . (G,DD)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,DD)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,DD)),F)) is set
K215(K266( the carrier of (G . (G,DD)),F),(LinearOperators ((G . (G,DD)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,DD)),F)
K213(K266( the carrier of (G . (G,DD)),F),(LinearOperators ((G . (G,DD)),F))) is Relation-like K7((LinearOperators ((G . (G,DD)),F)),(LinearOperators ((G . (G,DD)),F))) -defined LinearOperators ((G . (G,DD)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,DD)),F)),(LinearOperators ((G . (G,DD)),F))),(LinearOperators ((G . (G,DD)),F))))
K7((LinearOperators ((G . (G,DD)),F)),(LinearOperators ((G . (G,DD)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,DD)),F)),(LinearOperators ((G . (G,DD)),F))),(LinearOperators ((G . (G,DD)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,DD)),F)),(LinearOperators ((G . (G,DD)),F))),(LinearOperators ((G . (G,DD)),F)))) is set
K214(K266( the carrier of (G . (G,DD)),F),(LinearOperators ((G . (G,DD)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,DD)),F))) -defined LinearOperators ((G . (G,DD)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,DD)),F))),(LinearOperators ((G . (G,DD)),F))))
K7(REAL,(LinearOperators ((G . (G,DD)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,DD)),F))),(LinearOperators ((G . (G,DD)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,DD)),F))),(LinearOperators ((G . (G,DD)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,DD)),F)),K215(K266( the carrier of (G . (G,DD)),F),(LinearOperators ((G . (G,DD)),F))),K213(K266( the carrier of (G . (G,DD)),F),(LinearOperators ((G . (G,DD)),F))),K214(K266( the carrier of (G . (G,DD)),F),(LinearOperators ((G . (G,DD)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,DD)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,DD)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))) is Element of BoundedLinearOperators ((G . (G,DD)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))) -defined BoundedLinearOperators ((G . (G,DD)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))),(BoundedLinearOperators ((G . (G,DD)),F))))
K7((BoundedLinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))),(BoundedLinearOperators ((G . (G,DD)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))),(BoundedLinearOperators ((G . (G,DD)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,DD)),F))) -defined BoundedLinearOperators ((G . (G,DD)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,DD)),F))),(BoundedLinearOperators ((G . (G,DD)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,DD)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,DD)),F))),(BoundedLinearOperators ((G . (G,DD)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,DD)),F))),(BoundedLinearOperators ((G . (G,DD)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,DD)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,DD)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,DD)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,DD)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,DD)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,DD)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))),(BoundedLinearOperatorsNorm ((G . (G,DD)),F)) #) is strict NORMSTR
(G,F,DD,f,X) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F))
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)) is non empty set
(G,(G,DD),X) is non empty Relation-like the carrier of (G . (G,DD)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,DD)), the carrier of (product G)))
K7( the carrier of (G . (G,DD)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,DD)), the carrier of (product G))) is set
f * (G,(G,DD),X) is Relation-like the carrier of (G . (G,DD)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,DD)), the carrier of F))
(G,(G,DD)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,DD)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,DD))))
K7( the carrier of (product G), the carrier of (G . (G,DD))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,DD)))) is set
(G,(G,DD)) . X is Element of the carrier of (G . (G,DD))
diff ((f * (G,(G,DD),X)),((G,(G,DD)) . X)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F))
||.(G,F,DD,f,X).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)) -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 ((G . (G,DD)),F)),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)),REAL)) is non trivial V49() set
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)) . (G,F,DD,f,X) is V11() real ext-real Element of REAL
||.(G,F,DD,f,X).|| * ||.s.|| is V11() real ext-real non negative Element of REAL
a is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
(G,a) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . (G,a) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
R_NormSpace_of_BoundedLinearOperators ((G . (G,a)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,a)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,a)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,a)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,a)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,a)),F))
the carrier of (G . (G,a)) is non empty non trivial set
K266( the carrier of (G . (G,a)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,a)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,a)), the carrier of F
FuncZero ( the carrier of (G . (G,a)),F) is Relation-like the carrier of (G . (G,a)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,a)), the carrier of F)
the carrier of (G . (G,a)) --> (0. F) is non empty Relation-like the carrier of (G . (G,a)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,a)), the carrier of F))
K7( the carrier of (G . (G,a)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,a)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,a)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,a)), the carrier of F)),(Funcs ( the carrier of (G . (G,a)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,a)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,a)), the carrier of F)),(Funcs ( the carrier of (G . (G,a)), the carrier of F))),(Funcs ( the carrier of (G . (G,a)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,a)), the carrier of F)),(Funcs ( the carrier of (G . (G,a)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,a)), the carrier of F)),(Funcs ( the carrier of (G . (G,a)), the carrier of F))),(Funcs ( the carrier of (G . (G,a)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,a)), the carrier of F)),(Funcs ( the carrier of (G . (G,a)), the carrier of F))),(Funcs ( the carrier of (G . (G,a)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,a)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,a)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,a)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,a)), the carrier of F))),(Funcs ( the carrier of (G . (G,a)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,a)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,a)), the carrier of F))),(Funcs ( the carrier of (G . (G,a)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,a)), the carrier of F))),(Funcs ( the carrier of (G . (G,a)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,a)), the carrier of F)),(FuncZero ( the carrier of (G . (G,a)),F)),(FuncAdd ( the carrier of (G . (G,a)),F)),(FuncExtMult ( the carrier of (G . (G,a)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,a)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,a)),F)) is set
K215(K266( the carrier of (G . (G,a)),F),(LinearOperators ((G . (G,a)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,a)),F)
K213(K266( the carrier of (G . (G,a)),F),(LinearOperators ((G . (G,a)),F))) is Relation-like K7((LinearOperators ((G . (G,a)),F)),(LinearOperators ((G . (G,a)),F))) -defined LinearOperators ((G . (G,a)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,a)),F)),(LinearOperators ((G . (G,a)),F))),(LinearOperators ((G . (G,a)),F))))
K7((LinearOperators ((G . (G,a)),F)),(LinearOperators ((G . (G,a)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,a)),F)),(LinearOperators ((G . (G,a)),F))),(LinearOperators ((G . (G,a)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,a)),F)),(LinearOperators ((G . (G,a)),F))),(LinearOperators ((G . (G,a)),F)))) is set
K214(K266( the carrier of (G . (G,a)),F),(LinearOperators ((G . (G,a)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,a)),F))) -defined LinearOperators ((G . (G,a)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,a)),F))),(LinearOperators ((G . (G,a)),F))))
K7(REAL,(LinearOperators ((G . (G,a)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,a)),F))),(LinearOperators ((G . (G,a)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,a)),F))),(LinearOperators ((G . (G,a)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,a)),F)),K215(K266( the carrier of (G . (G,a)),F),(LinearOperators ((G . (G,a)),F))),K213(K266( the carrier of (G . (G,a)),F),(LinearOperators ((G . (G,a)),F))),K214(K266( the carrier of (G . (G,a)),F),(LinearOperators ((G . (G,a)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,a)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,a)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))) is Element of BoundedLinearOperators ((G . (G,a)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))) -defined BoundedLinearOperators ((G . (G,a)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))),(BoundedLinearOperators ((G . (G,a)),F))))
K7((BoundedLinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))),(BoundedLinearOperators ((G . (G,a)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))),(BoundedLinearOperators ((G . (G,a)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,a)),F))) -defined BoundedLinearOperators ((G . (G,a)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,a)),F))),(BoundedLinearOperators ((G . (G,a)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,a)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,a)),F))),(BoundedLinearOperators ((G . (G,a)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,a)),F))),(BoundedLinearOperators ((G . (G,a)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,a)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,a)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,a)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,a)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,a)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,a)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))),(BoundedLinearOperatorsNorm ((G . (G,a)),F)) #) is strict NORMSTR
(G,F,a,f,X) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,a)),F))
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,a)),F)) is non empty set
(G,(G,a),X) is non empty Relation-like the carrier of (G . (G,a)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,a)), the carrier of (product G)))
K7( the carrier of (G . (G,a)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,a)), the carrier of (product G))) is set
f * (G,(G,a),X) is Relation-like the carrier of (G . (G,a)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,a)), the carrier of F))
(G,(G,a)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,a)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,a))))
K7( the carrier of (product G), the carrier of (G . (G,a))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,a)))) is set
(G,(G,a)) . X is Element of the carrier of (G . (G,a))
diff ((f * (G,(G,a),X)),((G,(G,a)) . X)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,a)),F))
||.(G,F,a,f,X).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,a)),F)) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,a)),F)) -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 ((G . (G,a)),F)),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,a)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,a)),F)),REAL)) is non trivial V49() set
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,a)),F)) . (G,F,a,f,X) is V11() real ext-real Element of REAL
||.(G,F,a,f,X).|| * ||.s.|| is V11() real ext-real non negative Element of REAL
DD is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
a is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
y1 /. a is Element of the carrier of F
||.(y1 /. a).|| is V11() real ext-real non negative Element of REAL
the normF of F . (y1 /. a) is V11() real ext-real Element of REAL
t is V11() real ext-real Element of REAL
tm is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
y1 /. tm is Element of the carrier of F
||.(y1 /. tm).|| is V11() real ext-real non negative Element of REAL
the normF of F . (y1 /. tm) is V11() real ext-real Element of REAL
DD is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
dom DD is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
a is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
DD . a is V11() real ext-real Element of REAL
y1 /. a is Element of the carrier of F
||.(y1 /. a).|| is V11() real ext-real non negative Element of REAL
the normF of F . (y1 /. a) is V11() real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
y1 /. t is Element of the carrier of F
||.(y1 /. t).|| is V11() real ext-real non negative Element of REAL
the normF of F . (y1 /. t) is V11() real ext-real Element of REAL
len y1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
len DD is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
Sum DD is V11() real ext-real Element of REAL
(len G) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
REAL * is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like V49() FinSequence-like FinSubsequence-like countable Element of REAL * : len b₁ = len G } is set
len PP is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
a is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
DD . a is V11() real ext-real Element of REAL
y1 /. a is Element of the carrier of F
||.(y1 /. a).|| is V11() real ext-real non negative Element of REAL
the normF of F . (y1 /. a) is V11() real ext-real Element of REAL
y1 . a is set
(G,a) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . (G,a) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
(G,F,a,f,X) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,a)),F))
R_NormSpace_of_BoundedLinearOperators ((G . (G,a)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,a)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,a)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,a)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,a)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,a)),F))
the carrier of (G . (G,a)) is non empty non trivial set
K266( the carrier of (G . (G,a)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,a)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,a)), the carrier of F
FuncZero ( the carrier of (G . (G,a)),F) is Relation-like the carrier of (G . (G,a)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,a)), the carrier of F)
the carrier of (G . (G,a)) --> (0. F) is non empty Relation-like the carrier of (G . (G,a)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,a)), the carrier of F))
K7( the carrier of (G . (G,a)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,a)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,a)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,a)), the carrier of F)),(Funcs ( the carrier of (G . (G,a)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,a)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,a)), the carrier of F)),(Funcs ( the carrier of (G . (G,a)), the carrier of F))),(Funcs ( the carrier of (G . (G,a)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,a)), the carrier of F)),(Funcs ( the carrier of (G . (G,a)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,a)), the carrier of F)),(Funcs ( the carrier of (G . (G,a)), the carrier of F))),(Funcs ( the carrier of (G . (G,a)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,a)), the carrier of F)),(Funcs ( the carrier of (G . (G,a)), the carrier of F))),(Funcs ( the carrier of (G . (G,a)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,a)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,a)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,a)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,a)), the carrier of F))),(Funcs ( the carrier of (G . (G,a)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,a)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,a)), the carrier of F))),(Funcs ( the carrier of (G . (G,a)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,a)), the carrier of F))),(Funcs ( the carrier of (G . (G,a)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,a)), the carrier of F)),(FuncZero ( the carrier of (G . (G,a)),F)),(FuncAdd ( the carrier of (G . (G,a)),F)),(FuncExtMult ( the carrier of (G . (G,a)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,a)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,a)),F)) is set
K215(K266( the carrier of (G . (G,a)),F),(LinearOperators ((G . (G,a)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,a)),F)
K213(K266( the carrier of (G . (G,a)),F),(LinearOperators ((G . (G,a)),F))) is Relation-like K7((LinearOperators ((G . (G,a)),F)),(LinearOperators ((G . (G,a)),F))) -defined LinearOperators ((G . (G,a)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,a)),F)),(LinearOperators ((G . (G,a)),F))),(LinearOperators ((G . (G,a)),F))))
K7((LinearOperators ((G . (G,a)),F)),(LinearOperators ((G . (G,a)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,a)),F)),(LinearOperators ((G . (G,a)),F))),(LinearOperators ((G . (G,a)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,a)),F)),(LinearOperators ((G . (G,a)),F))),(LinearOperators ((G . (G,a)),F)))) is set
K214(K266( the carrier of (G . (G,a)),F),(LinearOperators ((G . (G,a)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,a)),F))) -defined LinearOperators ((G . (G,a)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,a)),F))),(LinearOperators ((G . (G,a)),F))))
K7(REAL,(LinearOperators ((G . (G,a)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,a)),F))),(LinearOperators ((G . (G,a)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,a)),F))),(LinearOperators ((G . (G,a)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,a)),F)),K215(K266( the carrier of (G . (G,a)),F),(LinearOperators ((G . (G,a)),F))),K213(K266( the carrier of (G . (G,a)),F),(LinearOperators ((G . (G,a)),F))),K214(K266( the carrier of (G . (G,a)),F),(LinearOperators ((G . (G,a)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,a)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,a)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))) is Element of BoundedLinearOperators ((G . (G,a)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))) -defined BoundedLinearOperators ((G . (G,a)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))),(BoundedLinearOperators ((G . (G,a)),F))))
K7((BoundedLinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))),(BoundedLinearOperators ((G . (G,a)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))),(BoundedLinearOperators ((G . (G,a)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,a)),F))) -defined BoundedLinearOperators ((G . (G,a)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,a)),F))),(BoundedLinearOperators ((G . (G,a)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,a)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,a)),F))),(BoundedLinearOperators ((G . (G,a)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,a)),F))),(BoundedLinearOperators ((G . (G,a)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,a)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,a)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,a)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,a)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,a)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,a)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))),(BoundedLinearOperatorsNorm ((G . (G,a)),F)) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,a)),F)) is non empty set
(G,(G,a),X) is non empty Relation-like the carrier of (G . (G,a)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,a)), the carrier of (product G)))
K7( the carrier of (G . (G,a)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,a)), the carrier of (product G))) is set
f * (G,(G,a),X) is Relation-like the carrier of (G . (G,a)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,a)), the carrier of F))
(G,(G,a)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,a)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,a))))
K7( the carrier of (product G), the carrier of (G . (G,a))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,a)))) is set
(G,(G,a)) . X is Element of the carrier of (G . (G,a))
diff ((f * (G,(G,a),X)),((G,(G,a)) . X)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,a)),F))
(G,(G,a)) . s is Element of the carrier of (G . (G,a))
(G,F,a,f,X) . ((G,(G,a)) . s) is Element of the carrier of F
||.((G,F,a,f,X) . ((G,(G,a)) . s)).|| is V11() real ext-real non negative Element of REAL
the normF of F . ((G,F,a,f,X) . ((G,(G,a)) . s)) is V11() real ext-real Element of REAL
t is non empty Relation-like the carrier of (G . (G,a)) -defined the carrier of F -valued Function-like total quasi_total V153(G . (G,a),F) V154(G . (G,a),F) Lipschitzian Element of K6(K7( the carrier of (G . (G,a)), the carrier of F))
t . ((G,(G,a)) . s) is Element of the carrier of F
||.(t . ((G,(G,a)) . s)).|| is V11() real ext-real non negative Element of REAL
the normF of F . (t . ((G,(G,a)) . s)) is V11() real ext-real Element of REAL
||.(G,F,a,f,X).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,a)),F)) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,a)),F)) -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 ((G . (G,a)),F)),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,a)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,a)),F)),REAL)) is non trivial V49() set
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,a)),F)) . (G,F,a,f,X) is V11() real ext-real Element of REAL
||.((G,(G,a)) . s).|| is V11() real ext-real non negative Element of REAL
the normF of (G . (G,a)) is non empty Relation-like the carrier of (G . (G,a)) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (G . (G,a)),REAL))
K7( the carrier of (G . (G,a)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (G . (G,a)),REAL)) is non trivial V49() set
the normF of (G . (G,a)) . ((G,(G,a)) . s) is V11() real ext-real Element of REAL
||.(G,F,a,f,X).|| * ||.((G,(G,a)) . s).|| is V11() real ext-real non negative Element of REAL
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
zeros G is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
addop G is Relation-like Function-like BinOps of carr G
[:(addop G):] is Relation-like K7((product (carr G)),(product (carr G))) -defined product (carr G) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((product (carr G)),(product (carr G))),(product (carr G))))
K7((product (carr G)),(product (carr G))) is Relation-like set
K7(K7((product (carr G)),(product (carr G))),(product (carr G))) is Relation-like set
K6(K7(K7((product (carr G)),(product (carr G))),(product (carr G)))) is set
multop G is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr G
[:(multop G):] is Relation-like K7(REAL,(product (carr G))) -defined product (carr G) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(product (carr G))),(product (carr G))))
K7(REAL,(product (carr G))) is non trivial Relation-like V49() set
K7(K7(REAL,(product (carr G))),(product (carr G))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(product (carr G))),(product (carr G)))) is non trivial V49() set
productnorm G is non empty Relation-like product (carr G) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((product (carr G)),REAL))
K7((product (carr G)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((product (carr G)),REAL)) is non trivial V49() set
NORMSTR(# (product (carr G)),(zeros G),[:(addop G):],[:(multop G):],(productnorm G) #) is strict NORMSTR
Sw1 is Element of the carrier of (G . (G,a))
tm is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
(G,tm,(G,a)) is Element of the carrier of (G . (G,a))
G . (G,a) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . (G,a)) is non empty set
||.(G,F,a,f,X).|| * ||.s.|| is V11() real ext-real non negative Element of REAL
PP . a is V11() real ext-real Element of REAL
Sum PP is V11() real ext-real Element of REAL
len y0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
a is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() len G -element FinSequence-like FinSubsequence-like countable Element of (len G) -tuples_on REAL
||.s.|| * a is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() len G -element FinSequence-like FinSubsequence-like countable Element of (len G) -tuples_on REAL
||.s.|| * y0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
dom (||.s.|| * y0) is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
t is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable Element of REAL *
len t is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
t is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
PP . t is V11() real ext-real Element of REAL
(G,t) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . (G,t) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
R_NormSpace_of_BoundedLinearOperators ((G . (G,t)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,t)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,t)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,t)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,t)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,t)),F))
the carrier of (G . (G,t)) is non empty non trivial set
K266( the carrier of (G . (G,t)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,t)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,t)), the carrier of F
FuncZero ( the carrier of (G . (G,t)),F) is Relation-like the carrier of (G . (G,t)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,t)), the carrier of F)
the carrier of (G . (G,t)) --> (0. F) is non empty Relation-like the carrier of (G . (G,t)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,t)), the carrier of F))
K7( the carrier of (G . (G,t)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,t)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,t)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,t)), the carrier of F)),(Funcs ( the carrier of (G . (G,t)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,t)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,t)), the carrier of F)),(Funcs ( the carrier of (G . (G,t)), the carrier of F))),(Funcs ( the carrier of (G . (G,t)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,t)), the carrier of F)),(Funcs ( the carrier of (G . (G,t)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,t)), the carrier of F)),(Funcs ( the carrier of (G . (G,t)), the carrier of F))),(Funcs ( the carrier of (G . (G,t)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,t)), the carrier of F)),(Funcs ( the carrier of (G . (G,t)), the carrier of F))),(Funcs ( the carrier of (G . (G,t)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,t)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,t)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,t)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,t)), the carrier of F))),(Funcs ( the carrier of (G . (G,t)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,t)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,t)), the carrier of F))),(Funcs ( the carrier of (G . (G,t)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,t)), the carrier of F))),(Funcs ( the carrier of (G . (G,t)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,t)), the carrier of F)),(FuncZero ( the carrier of (G . (G,t)),F)),(FuncAdd ( the carrier of (G . (G,t)),F)),(FuncExtMult ( the carrier of (G . (G,t)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,t)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,t)),F)) is set
K215(K266( the carrier of (G . (G,t)),F),(LinearOperators ((G . (G,t)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,t)),F)
K213(K266( the carrier of (G . (G,t)),F),(LinearOperators ((G . (G,t)),F))) is Relation-like K7((LinearOperators ((G . (G,t)),F)),(LinearOperators ((G . (G,t)),F))) -defined LinearOperators ((G . (G,t)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,t)),F)),(LinearOperators ((G . (G,t)),F))),(LinearOperators ((G . (G,t)),F))))
K7((LinearOperators ((G . (G,t)),F)),(LinearOperators ((G . (G,t)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,t)),F)),(LinearOperators ((G . (G,t)),F))),(LinearOperators ((G . (G,t)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,t)),F)),(LinearOperators ((G . (G,t)),F))),(LinearOperators ((G . (G,t)),F)))) is set
K214(K266( the carrier of (G . (G,t)),F),(LinearOperators ((G . (G,t)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,t)),F))) -defined LinearOperators ((G . (G,t)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,t)),F))),(LinearOperators ((G . (G,t)),F))))
K7(REAL,(LinearOperators ((G . (G,t)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,t)),F))),(LinearOperators ((G . (G,t)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,t)),F))),(LinearOperators ((G . (G,t)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,t)),F)),K215(K266( the carrier of (G . (G,t)),F),(LinearOperators ((G . (G,t)),F))),K213(K266( the carrier of (G . (G,t)),F),(LinearOperators ((G . (G,t)),F))),K214(K266( the carrier of (G . (G,t)),F),(LinearOperators ((G . (G,t)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,t)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,t)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,t)),F)),(BoundedLinearOperators ((G . (G,t)),F))) is Element of BoundedLinearOperators ((G . (G,t)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,t)),F)),(BoundedLinearOperators ((G . (G,t)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,t)),F)),(BoundedLinearOperators ((G . (G,t)),F))) -defined BoundedLinearOperators ((G . (G,t)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,t)),F)),(BoundedLinearOperators ((G . (G,t)),F))),(BoundedLinearOperators ((G . (G,t)),F))))
K7((BoundedLinearOperators ((G . (G,t)),F)),(BoundedLinearOperators ((G . (G,t)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,t)),F)),(BoundedLinearOperators ((G . (G,t)),F))),(BoundedLinearOperators ((G . (G,t)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,t)),F)),(BoundedLinearOperators ((G . (G,t)),F))),(BoundedLinearOperators ((G . (G,t)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,t)),F)),(BoundedLinearOperators ((G . (G,t)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,t)),F))) -defined BoundedLinearOperators ((G . (G,t)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,t)),F))),(BoundedLinearOperators ((G . (G,t)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,t)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,t)),F))),(BoundedLinearOperators ((G . (G,t)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,t)),F))),(BoundedLinearOperators ((G . (G,t)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,t)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,t)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,t)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,t)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,t)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,t)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,t)),F)),(BoundedLinearOperators ((G . (G,t)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,t)),F)),(BoundedLinearOperators ((G . (G,t)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,t)),F)),(BoundedLinearOperators ((G . (G,t)),F))),(BoundedLinearOperatorsNorm ((G . (G,t)),F)) #) is strict NORMSTR
(G,F,t,f,X) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,t)),F))
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,t)),F)) is non empty set
(G,(G,t),X) is non empty Relation-like the carrier of (G . (G,t)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,t)), the carrier of (product G)))
K7( the carrier of (G . (G,t)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,t)), the carrier of (product G))) is set
f * (G,(G,t),X) is Relation-like the carrier of (G . (G,t)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,t)), the carrier of F))
(G,(G,t)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,t)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,t))))
K7( the carrier of (product G), the carrier of (G . (G,t))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,t)))) is set
(G,(G,t)) . X is Element of the carrier of (G . (G,t))
diff ((f * (G,(G,t),X)),((G,(G,t)) . X)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,t)),F))
||.(G,F,t,f,X).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,t)),F)) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,t)),F)) -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 ((G . (G,t)),F)),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,t)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,t)),F)),REAL)) is non trivial V49() set
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,t)),F)) . (G,F,t,f,X) is V11() real ext-real Element of REAL
||.(G,F,t,f,X).|| * ||.s.|| is V11() real ext-real non negative Element of REAL
y0 . t is V11() real ext-real Element of REAL
||.s.|| * (y0 . t) is V11() real ext-real Element of REAL
(||.s.|| * y0) . t is V11() real ext-real Element of REAL
s is non empty Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like total quasi_total V153( product G,F) V154( product G,F) Lipschitzian Element of K6(K7( the carrier of (product G), the carrier of F))
y1 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
s . y1 is Element of the carrier of F
G is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of G is non empty set
F is Relation-like NAT -defined the carrier of G -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of G
len F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
f is Relation-like NAT -defined the carrier of G -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of G
len f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
(len f) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
dom f is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
F /. 1 is Element of the carrier of G
F /. (len F) is Element of the carrier of G
(F /. 1) - (F /. (len F)) is Element of the carrier of G
- (F /. (len F)) is Element of the carrier of G
(F /. 1) + (- (F /. (len F))) is Element of the carrier of G
the addF of G is Relation-like K7( the carrier of G, the carrier of G) -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of G, the carrier of G), the carrier of G))
K7( the carrier of G, the carrier of G) is Relation-like set
K7(K7( the carrier of G, the carrier of G), the carrier of G) is Relation-like set
K6(K7(K7( the carrier of G, the carrier of G), the carrier of G)) is set
K462( the carrier of G, the addF of G,(F /. 1),(- (F /. (len F)))) is Element of the carrier of G
Sum f is Element of the carrier of G
K7(NAT, the carrier of G) is non trivial Relation-like V49() set
K6(K7(NAT, the carrier of G)) is non trivial V49() set
0. G is V68(G) Element of the carrier of G
the ZeroF of G is Element of the carrier of G
X is non empty Relation-like NAT -defined the carrier of G -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of G))
X . (len f) is Element of the carrier of G
X . 0 is Element of the carrier of G
X . 1 is Element of the carrier of G
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
F /. (1 + 1) is Element of the carrier of G
(F /. 1) - (F /. (1 + 1)) is Element of the carrier of G
- (F /. (1 + 1)) is Element of the carrier of G
(F /. 1) + (- (F /. (1 + 1))) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. 1),(- (F /. (1 + 1)))) is Element of the carrier of G
Seg (len f) is V49() len f -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= len f ) } is set
f /. 1 is Element of the carrier of G
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
m + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
f . (m + 1) is set
X . (m + 1) is Element of the carrier of G
(X . 0) + (f /. 1) is Element of the carrier of G
K462( the carrier of G, the addF of G,(X . 0),(f /. 1)) is Element of the carrier of G
X . 1 is Element of the carrier of G
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
X . m is Element of the carrier of G
m + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
F /. (m + 1) is Element of the carrier of G
(F /. 1) - (F /. (m + 1)) is Element of the carrier of G
- (F /. (m + 1)) is Element of the carrier of G
(F /. 1) + (- (F /. (m + 1))) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. 1),(- (F /. (m + 1)))) is Element of the carrier of G
X . (m + 1) is Element of the carrier of G
(m + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
F /. ((m + 1) + 1) is Element of the carrier of G
(F /. 1) - (F /. ((m + 1) + 1)) is Element of the carrier of G
- (F /. ((m + 1) + 1)) is Element of the carrier of G
(F /. 1) + (- (F /. ((m + 1) + 1))) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. 1),(- (F /. ((m + 1) + 1)))) is Element of the carrier of G
f . (m + 1) is set
f /. (m + 1) is Element of the carrier of G
X . (m + 1) is Element of the carrier of G
(X . m) + (f /. (m + 1)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(X . m),(f /. (m + 1))) is Element of the carrier of G
(F /. (m + 1)) - (F /. ((m + 1) + 1)) is Element of the carrier of G
(F /. (m + 1)) + (- (F /. ((m + 1) + 1))) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. (m + 1)),(- (F /. ((m + 1) + 1)))) is Element of the carrier of G
(X . m) + ((F /. (m + 1)) - (F /. ((m + 1) + 1))) is Element of the carrier of G
K462( the carrier of G, the addF of G,(X . m),((F /. (m + 1)) - (F /. ((m + 1) + 1)))) is Element of the carrier of G
((F /. 1) - (F /. (m + 1))) + (F /. (m + 1)) is Element of the carrier of G
K462( the carrier of G, the addF of G,((F /. 1) - (F /. (m + 1))),(F /. (m + 1))) is Element of the carrier of G
(((F /. 1) - (F /. (m + 1))) + (F /. (m + 1))) - (F /. ((m + 1) + 1)) is Element of the carrier of G
(((F /. 1) - (F /. (m + 1))) + (F /. (m + 1))) + (- (F /. ((m + 1) + 1))) is Element of the carrier of G
K462( the carrier of G, the addF of G,(((F /. 1) - (F /. (m + 1))) + (F /. (m + 1))),(- (F /. ((m + 1) + 1)))) is Element of the carrier of G
(F /. (m + 1)) - (F /. (m + 1)) is Element of the carrier of G
(F /. (m + 1)) + (- (F /. (m + 1))) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. (m + 1)),(- (F /. (m + 1)))) is Element of the carrier of G
(F /. 1) - ((F /. (m + 1)) - (F /. (m + 1))) is Element of the carrier of G
- ((F /. (m + 1)) - (F /. (m + 1))) is Element of the carrier of G
(F /. 1) + (- ((F /. (m + 1)) - (F /. (m + 1)))) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. 1),(- ((F /. (m + 1)) - (F /. (m + 1))))) is Element of the carrier of G
((F /. 1) - ((F /. (m + 1)) - (F /. (m + 1)))) - (F /. ((m + 1) + 1)) is Element of the carrier of G
((F /. 1) - ((F /. (m + 1)) - (F /. (m + 1)))) + (- (F /. ((m + 1) + 1))) is Element of the carrier of G
K462( the carrier of G, the addF of G,((F /. 1) - ((F /. (m + 1)) - (F /. (m + 1)))),(- (F /. ((m + 1) + 1)))) is Element of the carrier of G
(F /. 1) - (0. G) is Element of the carrier of G
- (0. G) is Element of the carrier of G
(F /. 1) + (- (0. G)) is Element of the carrier of G
K462( the carrier of G, the addF of G,(F /. 1),(- (0. G))) is Element of the carrier of G
((F /. 1) - (0. G)) - (F /. ((m + 1) + 1)) is Element of the carrier of G
((F /. 1) - (0. G)) + (- (F /. ((m + 1) + 1))) is Element of the carrier of G
K462( the carrier of G, the addF of G,((F /. 1) - (0. G)),(- (F /. ((m + 1) + 1)))) is Element of the carrier of G
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
F is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
f is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
X is set
f | X is Relation-like NAT -defined X -defined NAT -defined Function-like carr G -compatible Element of sproduct (carr G)
sproduct (carr G) is non empty functional set
F +* (f | X) is Relation-like NAT -defined Function-like carr G -compatible set
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
0. (product G) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like V68( product G) countable Element of the carrier of (product G)
the ZeroF of (product G) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
F is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.f.|| is V11() real ext-real non negative Element of REAL
the normF of (product G) is non empty Relation-like the carrier of (product G) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (product G),REAL))
K7( the carrier of (product G),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (product G),REAL)) is non trivial V49() set
the normF of (product G) . f is V11() real ext-real Element of REAL
||.F.|| is V11() real ext-real non negative Element of REAL
the normF of (product G) . F is V11() real ext-real Element of REAL
X is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
m is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
i is set
m | i is Relation-like NAT -defined i -defined NAT -defined Function-like carr G -compatible Element of sproduct (carr G)
sproduct (carr G) is non empty functional set
X +* (m | i) is Relation-like NAT -defined Function-like carr G -compatible set
productnorm G is non empty Relation-like product (carr G) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((product (carr G)),REAL))
K7((product (carr G)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((product (carr G)),REAL)) is non trivial V49() set
(productnorm G) . f is V11() real ext-real Element of REAL
i0 is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
normsequence (G,i0) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() len G -element FinSequence-like FinSubsequence-like countable Element of REAL (len G)
REAL (len G) is non empty functional FinSequence-membered FinSequenceSet of REAL
(len G) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
REAL * is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like V49() FinSequence-like FinSubsequence-like countable Element of REAL * : len b₁ = len G } is set
|.(normsequence (G,i0)).| is V11() real ext-real non negative Element of REAL
sqr (normsequence (G,i0)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
Sum (sqr (normsequence (G,i0))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (normsequence (G,i0)))) is V11() real ext-real Element of REAL
(productnorm G) . F is V11() real ext-real Element of REAL
normsequence (G,m) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() len G -element FinSequence-like FinSubsequence-like countable Element of REAL (len G)
|.(normsequence (G,m)).| is V11() real ext-real non negative Element of REAL
sqr (normsequence (G,m)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
Sum (sqr (normsequence (G,m))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (normsequence (G,m)))) is V11() real ext-real Element of REAL
r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() len G -element FinSequence-like FinSubsequence-like countable FinSequence of REAL
len r is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
y0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() len G -element FinSequence-like FinSubsequence-like countable FinSequence of REAL
len y0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
Seg (len y0) is V49() len y0 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= len y0 ) } is set
s is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
y0 . s is V11() real ext-real Element of REAL
r . s is V11() real ext-real Element of REAL
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
PP is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable set
dom PP is V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
Seg (len G) is non empty V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= len G ) } is set
dom m is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below Element of K6(NAT)
y1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . y1 is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . y1) is non empty non trivial set
(G,i0,y1) is Element of the carrier of (G . y1)
G . y1 is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . y1) is non empty set
(G,m,y1) is Element of the carrier of (G . y1)
the normF of (G . y1) is non empty Relation-like the carrier of (G . y1) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (G . y1),REAL))
K7( the carrier of (G . y1),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (G . y1),REAL)) is non trivial V49() set
the normF of (G . y1) . (G,m,y1) is V11() real ext-real Element of REAL
DD is Element of the carrier of (G . y1)
||.DD.|| is V11() real ext-real non negative Element of REAL
the normF of (G . y1) . DD is V11() real ext-real Element of REAL
a is Element of the carrier of (G . y1)
||.a.|| is V11() real ext-real non negative Element of REAL
the normF of (G . y1) . a is V11() real ext-real Element of REAL
dom (m | i) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(i)
K6(i) is set
(m | i) . y1 is set
dom (m | i) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below Element of K6(i)
K6(i) is set
(G,X,y1) is Element of the carrier of (G . y1)
0. (G . y1) is V68(G . y1) Element of the carrier of (G . y1)
the ZeroF of (G . y1) is Element of the carrier of (G . y1)
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
0. (product G) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like V68( product G) countable Element of the carrier of (product G)
the ZeroF of (product G) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(len G) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of F is non empty non trivial set
K7( the carrier of (product G), the carrier of F) is Relation-like set
K6(K7( the carrier of (product G), the carrier of F)) is set
f is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of F))
m is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
X is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.m.|| is V11() real ext-real non negative Element of REAL
the normF of (product G) is non empty Relation-like the carrier of (product G) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (product G),REAL))
K7( the carrier of (product G),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (product G),REAL)) is non trivial V49() set
the normF of (product G) . m is V11() real ext-real Element of REAL
X + m is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
the addF of (product G) is Relation-like K7( the carrier of (product G), the carrier of (product G)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)))
K7( the carrier of (product G), the carrier of (product G)) is Relation-like set
K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)) is Relation-like set
K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G))) is set
K462( the carrier of (product G), the addF of (product G),X,m) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f /. (X + m) is Element of the carrier of F
f /. X is Element of the carrier of F
(f /. (X + m)) - (f /. X) is Element of the carrier of F
- (f /. X) is Element of the carrier of F
(f /. (X + m)) + (- (f /. X)) is Element of the carrier of F
the addF of F is Relation-like K7( the carrier of F, the carrier of F) -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of F, the carrier of F), the carrier of F))
K7( the carrier of F, the carrier of F) is Relation-like set
K7(K7( the carrier of F, the carrier of F), the carrier of F) is Relation-like set
K6(K7(K7( the carrier of F, the carrier of F), the carrier of F)) is set
K462( the carrier of F, the addF of F,(f /. (X + m)),(- (f /. X))) is Element of the carrier of F
r is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable set
len r is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
dom r is V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
s is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable set
len s is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
dom s is V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
i0 is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
y0 is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
Seg ((len G) + 1) is non empty V49() (len G) + 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= (len G) + 1 ) } is set
y1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
((len G) + 1) -' y1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
Seg (((len G) + 1) -' y1) is V49() ((len G) + 1) -' y1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= ((len G) + 1) -' y1 ) } is set
y0 | (Seg (((len G) + 1) -' y1)) is Relation-like NAT -defined Seg (((len G) + 1) -' y1) -defined NAT -defined Function-like carr G -compatible V49() countable Element of sproduct (carr G)
sproduct (carr G) is non empty functional set
i0 +* (y0 | (Seg (((len G) + 1) -' y1))) is Relation-like NAT -defined Function-like carr G -compatible set
y1 is Relation-like NAT -defined the carrier of (product G) -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of (product G)
dom y1 is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
PP is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
y1 /. PP is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
y1 . PP is set
((len G) + 1) -' PP is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
Seg (((len G) + 1) -' PP) is V49() ((len G) + 1) -' PP -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= ((len G) + 1) -' PP ) } is set
y0 | (Seg (((len G) + 1) -' PP)) is Relation-like NAT -defined Seg (((len G) + 1) -' PP) -defined NAT -defined Function-like carr G -compatible V49() countable Element of sproduct (carr G)
sproduct (carr G) is non empty functional set
i0 +* (y0 | (Seg (((len G) + 1) -' PP))) is Relation-like NAT -defined Function-like carr G -compatible set
PP is Relation-like NAT -defined the carrier of F -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of F
len PP is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
dom PP is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
DD is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
PP /. DD is Element of the carrier of F
PP . DD is set
y1 /. DD is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
X + (y1 /. DD) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),X,(y1 /. DD)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f /. (X + (y1 /. DD)) is Element of the carrier of F
DD is Relation-like NAT -defined the carrier of F -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of F
len DD is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
dom DD is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
a is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
DD /. a is Element of the carrier of F
DD . a is set
PP /. a is Element of the carrier of F
a + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
PP /. (a + 1) is Element of the carrier of F
(PP /. a) - (PP /. (a + 1)) is Element of the carrier of F
- (PP /. (a + 1)) is Element of the carrier of F
(PP /. a) + (- (PP /. (a + 1))) is Element of the carrier of F
K462( the carrier of F, the addF of F,(PP /. a),(- (PP /. (a + 1)))) is Element of the carrier of F
((len G) + 1) -' 1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
((len G) + 1) - 1 is V11() real ext-real V132() V160() Element of INT
K38(1) is non empty V11() real ext-real non positive negative set
K36(((len G) + 1),K38(1)) is V11() real ext-real set
y1 /. 1 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
Seg (((len G) + 1) -' 1) is V49() ((len G) + 1) -' 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= ((len G) + 1) -' 1 ) } is set
y0 | (Seg (((len G) + 1) -' 1)) is Relation-like NAT -defined Seg (((len G) + 1) -' 1) -defined NAT -defined Function-like carr G -compatible V49() countable Element of sproduct (carr G)
i0 +* (y0 | (Seg (((len G) + 1) -' 1))) is Relation-like NAT -defined Function-like carr G -compatible set
y0 | (dom G) is Relation-like NAT -defined dom G -defined NAT -defined Function-like carr G -compatible V49() countable Element of sproduct (carr G)
i0 +* (y0 | (dom G)) is Relation-like NAT -defined Function-like carr G -compatible set
i0 +* y0 is Relation-like NAT -defined Function-like carr G -compatible set
((len G) + 1) -' (len PP) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
((len G) + 1) - (len PP) is V11() real ext-real V132() V160() Element of INT
K38((len PP)) is V11() real ext-real non positive set
K36(((len G) + 1),K38((len PP))) is V11() real ext-real set
y1 /. (len PP) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
Seg 0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional complex-valued ext-real-valued real-valued natural-valued V49() V50() V53() cardinal 0 -element {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V134() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V167() Cardinal-yielding countable bounded_below bounded_above real-bounded V251() Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= 0 ) } is set
y0 | (Seg 0) is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined Seg 0 -defined NAT -defined RAT -valued Function-like one-to-one constant functional carr G -compatible complex-valued ext-real-valued real-valued natural-valued V49() V50() V53() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Function-yielding V134() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V167() Cardinal-yielding countable bounded_below bounded_above real-bounded V251() Element of sproduct (carr G)
i0 +* (y0 | (Seg 0)) is Relation-like NAT -defined Function-like carr G -compatible set
i0 +* {} is Relation-like NAT -defined Function-like set
PP /. 1 is Element of the carrier of F
PP /. (len PP) is Element of the carrier of F
X + (y1 /. (len PP)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),X,(y1 /. (len PP))) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f /. (X + (y1 /. (len PP))) is Element of the carrier of F
len y1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
Sum DD is Element of the carrier of F
t is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
Seg (len G) is non empty V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= len G ) } is set
t + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
PP /. t is Element of the carrier of F
y1 /. t is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
X + (y1 /. t) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),X,(y1 /. t)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f /. (X + (y1 /. t)) is Element of the carrier of F
PP /. (t + 1) is Element of the carrier of F
y1 /. (t + 1) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
X + (y1 /. (t + 1)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),X,(y1 /. (t + 1))) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f /. (X + (y1 /. (t + 1))) is Element of the carrier of F
DD /. t is Element of the carrier of F
(f /. (X + (y1 /. t))) - (f /. (X + (y1 /. (t + 1)))) is Element of the carrier of F
- (f /. (X + (y1 /. (t + 1)))) is Element of the carrier of F
(f /. (X + (y1 /. t))) + (- (f /. (X + (y1 /. (t + 1))))) is Element of the carrier of F
K462( the carrier of F, the addF of F,(f /. (X + (y1 /. t))),(- (f /. (X + (y1 /. (t + 1)))))) is Element of the carrier of F
t is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
y1 /. t is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
tm is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((len G) + 1) -' t is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
Seg (((len G) + 1) -' t) is V49() ((len G) + 1) -' t -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= ((len G) + 1) -' t ) } is set
y0 | (Seg (((len G) + 1) -' t)) is Relation-like NAT -defined Seg (((len G) + 1) -' t) -defined NAT -defined Function-like carr G -compatible V49() countable Element of sproduct (carr G)
i0 +* (y0 | (Seg (((len G) + 1) -' t))) is Relation-like NAT -defined Function-like carr G -compatible set
||.tm.|| is V11() real ext-real non negative Element of REAL
the normF of (product G) . tm is V11() real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
y1 /. t is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((len G) + 1) -' t is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
Seg (((len G) + 1) -' t) is V49() ((len G) + 1) -' t -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= ((len G) + 1) -' t ) } is set
y0 | (Seg (((len G) + 1) -' t)) is Relation-like NAT -defined Seg (((len G) + 1) -' t) -defined NAT -defined Function-like carr G -compatible V49() countable Element of sproduct (carr G)
i0 +* (y0 | (Seg (((len G) + 1) -' t))) is Relation-like NAT -defined Function-like carr G -compatible set
tm is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
DD /. tm is Element of the carrier of F
y1 /. tm is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
X + (y1 /. tm) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),X,(y1 /. tm)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f /. (X + (y1 /. tm)) is Element of the carrier of F
tm + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
y1 /. (tm + 1) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
X + (y1 /. (tm + 1)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),X,(y1 /. (tm + 1))) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f /. (X + (y1 /. (tm + 1))) is Element of the carrier of F
(f /. (X + (y1 /. tm))) - (f /. (X + (y1 /. (tm + 1)))) is Element of the carrier of F
- (f /. (X + (y1 /. (tm + 1)))) is Element of the carrier of F
(f /. (X + (y1 /. tm))) + (- (f /. (X + (y1 /. (tm + 1))))) is Element of the carrier of F
K462( the carrier of F, the addF of F,(f /. (X + (y1 /. tm))),(- (f /. (X + (y1 /. (tm + 1)))))) is Element of the carrier of F
Sw1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
y1 /. Sw1 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
w2 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.w2.|| is V11() real ext-real non negative Element of REAL
the normF of (product G) . w2 is V11() real ext-real Element of REAL
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . F) is non empty non trivial set
(G,F) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . F) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . F)))
K7( the carrier of (product G), the carrier of (G . F)) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . F))) is set
X is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,F,f) is non empty Relation-like the carrier of (G . F) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . F), the carrier of (product G)))
K7( the carrier of (G . F), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . F), the carrier of (product G))) is set
(G,F) . X is Element of the carrier of (G . F)
m is Element of the carrier of (G . F)
(G,F,f) . m is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
Replace (F,m,) is Relation-like Function-like set
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
dom (carr G) is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
i is Relation-like Function-like set
dom i is set
dom f is V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
(Replace (F,m,)) . F is set
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . F) is non empty non trivial set
(G,F) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . F) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . F)))
K7( the carrier of (product G), the carrier of (G . F)) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . F))) is set
f is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,F) . f is Element of the carrier of (G . F)
(G,F,f) is non empty Relation-like the carrier of (G . F) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . F), the carrier of (product G)))
K7( the carrier of (G . F), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . F), the carrier of (product G))) is set
X is Element of the carrier of (G . F)
(G,F,f) . X is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
m is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable set
dom m is V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
Seg (len G) is non empty V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= len G ) } is set
i is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable set
dom i is V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
i0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
m . i0 is set
i . i0 is set
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
zeros G is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
addop G is Relation-like Function-like BinOps of carr G
[:(addop G):] is Relation-like K7((product (carr G)),(product (carr G))) -defined product (carr G) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((product (carr G)),(product (carr G))),(product (carr G))))
K7((product (carr G)),(product (carr G))) is Relation-like set
K7(K7((product (carr G)),(product (carr G))),(product (carr G))) is Relation-like set
K6(K7(K7((product (carr G)),(product (carr G))),(product (carr G)))) is set
multop G is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr G
[:(multop G):] is Relation-like K7(REAL,(product (carr G))) -defined product (carr G) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(product (carr G))),(product (carr G))))
K7(REAL,(product (carr G))) is non trivial Relation-like V49() set
K7(K7(REAL,(product (carr G))),(product (carr G))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(product (carr G))),(product (carr G)))) is non trivial V49() set
productnorm G is non empty Relation-like product (carr G) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((product (carr G)),REAL))
K7((product (carr G)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((product (carr G)),REAL)) is non trivial V49() set
NORMSTR(# (product (carr G)),(zeros G),[:(addop G):],[:(multop G):],(productnorm G) #) is strict NORMSTR
f . F is set
Replace (F,X,) is Relation-like Function-like set
(Replace (F,X,)) . i0 is set
Replace (F,X,) is Relation-like Function-like set
(Replace (F,X,)) . i0 is set
f . i0 is set
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . F) is non empty non trivial set
X is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,F,f) is non empty Relation-like the carrier of (G . F) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . F), the carrier of (product G)))
K7( the carrier of (G . F), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . F), the carrier of (product G))) is set
(G,F,X) is non empty Relation-like the carrier of (G . F) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . F), the carrier of (product G)))
m is Element of the carrier of (G . F)
(G,F,f) . m is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
i is Element of the carrier of (G . F)
(G,F,f) . i is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,F,X) . i is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
Replace (F,i,) is Relation-like Function-like set
Replace (F,i,) is Relation-like Function-like set
i0 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable set
dom i0 is V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
Seg (len G) is non empty V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= len G ) } is set
y0 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable set
dom y0 is V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
r is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
i0 . r is set
y0 . r is set
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
dom (carr G) is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
s is Relation-like Function-like set
dom s is set
y1 is Relation-like Function-like set
dom y1 is set
dom X is V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
dom f is V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
(Replace (F,i,)) . r is set
(Replace (F,i,)) . r is set
X . r is set
f . r is set
Replace (F,m,) is Relation-like Function-like set
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . F) is non empty non trivial set
f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . f is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . f) is non empty non trivial set
(G,f) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . f) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . f)))
K7( the carrier of (product G), the carrier of (G . f)) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . f))) is set
m is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
X is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,F,X) is non empty Relation-like the carrier of (G . F) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . F), the carrier of (product G)))
K7( the carrier of (G . F), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . F), the carrier of (product G))) is set
(G,f) . X is Element of the carrier of (G . f)
(G,f) . m is Element of the carrier of (G . f)
i is Element of the carrier of (G . F)
(G,F,X) . i is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
Replace (F,i,) is Relation-like Function-like set
F .--> i is trivial Relation-like dom G -defined {F} -defined the carrier of (G . F) -valued Function-like one-to-one constant V49() countable set
{F} is non empty trivial V49() V53() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded set
i0 is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
(G,i0,f) is Element of the carrier of (G . f)
G . f is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . f) is non empty set
X . f is set
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of F is non empty non trivial set
K7( the carrier of (product G), the carrier of F) is Relation-like set
K6(K7( the carrier of (product G), the carrier of F)) is set
f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . f is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . f) is non empty non trivial set
K7( the carrier of (G . f), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . f), the carrier of F)) is set
(G,f) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . f) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . f)))
K7( the carrier of (product G), the carrier of (G . f)) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . f))) is set
X is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,f) . X is Element of the carrier of (G . f)
(G,f,X) is non empty Relation-like the carrier of (G . f) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . f), the carrier of (product G)))
K7( the carrier of (G . f), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . f), the carrier of (product G))) is set
m is Element of the carrier of (G . f)
i is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of F))
i * (G,f,X) is Relation-like the carrier of (G . f) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . f), the carrier of F))
(G,F,f,i,X) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F))
(G,f) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . (G,f) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,f)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,f)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,f)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,f)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,f)),F))
the carrier of (G . (G,f)) is non empty non trivial set
K266( the carrier of (G . (G,f)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,f)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,f)), the carrier of F
FuncZero ( the carrier of (G . (G,f)),F) is Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,f)), the carrier of F)
0. F is V68(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
the carrier of (G . (G,f)) --> (0. F) is non empty Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,f)), the carrier of F))
K7( the carrier of (G . (G,f)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,f)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,f)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,f)), the carrier of F)),(Funcs ( the carrier of (G . (G,f)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,f)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,f)), the carrier of F)),(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,f)), the carrier of F)),(Funcs ( the carrier of (G . (G,f)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,f)), the carrier of F)),(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,f)), the carrier of F)),(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,f)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,f)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,f)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,f)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,f)), the carrier of F))),(Funcs ( the carrier of (G . (G,f)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,f)), the carrier of F)),(FuncZero ( the carrier of (G . (G,f)),F)),(FuncAdd ( the carrier of (G . (G,f)),F)),(FuncExtMult ( the carrier of (G . (G,f)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,f)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,f)),F)) is set
K215(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,f)),F)
K213(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))) is Relation-like K7((LinearOperators ((G . (G,f)),F)),(LinearOperators ((G . (G,f)),F))) -defined LinearOperators ((G . (G,f)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,f)),F)),(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F))))
K7((LinearOperators ((G . (G,f)),F)),(LinearOperators ((G . (G,f)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,f)),F)),(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,f)),F)),(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F)))) is set
K214(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,f)),F))) -defined LinearOperators ((G . (G,f)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F))))
K7(REAL,(LinearOperators ((G . (G,f)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,f)),F))),(LinearOperators ((G . (G,f)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,f)),F)),K215(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))),K213(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))),K214(K266( the carrier of (G . (G,f)),F),(LinearOperators ((G . (G,f)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,f)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,f)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))) is Element of BoundedLinearOperators ((G . (G,f)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))) -defined BoundedLinearOperators ((G . (G,f)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F))))
K7((BoundedLinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,f)),F))) -defined BoundedLinearOperators ((G . (G,f)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,f)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperators ((G . (G,f)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,f)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,f)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,f)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,f)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,f)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,f)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,f)),F)),(BoundedLinearOperators ((G . (G,f)),F))),(BoundedLinearOperatorsNorm ((G . (G,f)),F)) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F)) is non empty set
(G,(G,f),X) is non empty Relation-like the carrier of (G . (G,f)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,f)), the carrier of (product G)))
K7( the carrier of (G . (G,f)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,f)), the carrier of (product G))) is set
i * (G,(G,f),X) is Relation-like the carrier of (G . (G,f)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,f)), the carrier of F))
(G,(G,f)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,f)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,f))))
K7( the carrier of (product G), the carrier of (G . (G,f))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,f)))) is set
(G,(G,f)) . X is Element of the carrier of (G . (G,f))
diff ((i * (G,(G,f),X)),((G,(G,f)) . X)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,f)),F))
i0 is Relation-like the carrier of (G . f) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . f), the carrier of F))
diff (i0,m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . f),F))
R_NormSpace_of_BoundedLinearOperators ((G . f),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . f),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . f),F)))
R_VectorSpace_of_LinearOperators ((G . f),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . f),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . f),F))
K266( the carrier of (G . f),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . f), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . f), the carrier of F
FuncZero ( the carrier of (G . f),F) is Relation-like the carrier of (G . f) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . f), the carrier of F)
the carrier of (G . f) --> (0. F) is non empty Relation-like the carrier of (G . f) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . f), the carrier of F))
FuncAdd ( the carrier of (G . f),F) is Relation-like K7((Funcs ( the carrier of (G . f), the carrier of F)),(Funcs ( the carrier of (G . f), the carrier of F))) -defined Funcs ( the carrier of (G . f), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . f), the carrier of F)),(Funcs ( the carrier of (G . f), the carrier of F))),(Funcs ( the carrier of (G . f), the carrier of F))))
K7((Funcs ( the carrier of (G . f), the carrier of F)),(Funcs ( the carrier of (G . f), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . f), the carrier of F)),(Funcs ( the carrier of (G . f), the carrier of F))),(Funcs ( the carrier of (G . f), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . f), the carrier of F)),(Funcs ( the carrier of (G . f), the carrier of F))),(Funcs ( the carrier of (G . f), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . f),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . f), the carrier of F))) -defined Funcs ( the carrier of (G . f), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . f), the carrier of F))),(Funcs ( the carrier of (G . f), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . f), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . f), the carrier of F))),(Funcs ( the carrier of (G . f), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . f), the carrier of F))),(Funcs ( the carrier of (G . f), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . f), the carrier of F)),(FuncZero ( the carrier of (G . f),F)),(FuncAdd ( the carrier of (G . f),F)),(FuncExtMult ( the carrier of (G . f),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . f),F) is non empty set
K6( the carrier of K266( the carrier of (G . f),F)) is set
K215(K266( the carrier of (G . f),F),(LinearOperators ((G . f),F))) is Relation-like Function-like Element of LinearOperators ((G . f),F)
K213(K266( the carrier of (G . f),F),(LinearOperators ((G . f),F))) is Relation-like K7((LinearOperators ((G . f),F)),(LinearOperators ((G . f),F))) -defined LinearOperators ((G . f),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . f),F)),(LinearOperators ((G . f),F))),(LinearOperators ((G . f),F))))
K7((LinearOperators ((G . f),F)),(LinearOperators ((G . f),F))) is Relation-like set
K7(K7((LinearOperators ((G . f),F)),(LinearOperators ((G . f),F))),(LinearOperators ((G . f),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . f),F)),(LinearOperators ((G . f),F))),(LinearOperators ((G . f),F)))) is set
K214(K266( the carrier of (G . f),F),(LinearOperators ((G . f),F))) is Relation-like K7(REAL,(LinearOperators ((G . f),F))) -defined LinearOperators ((G . f),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . f),F))),(LinearOperators ((G . f),F))))
K7(REAL,(LinearOperators ((G . f),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . f),F))),(LinearOperators ((G . f),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . f),F))),(LinearOperators ((G . f),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . f),F)),K215(K266( the carrier of (G . f),F),(LinearOperators ((G . f),F))),K213(K266( the carrier of (G . f),F),(LinearOperators ((G . f),F))),K214(K266( the carrier of (G . f),F),(LinearOperators ((G . f),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . f),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . f),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . f),F)),(BoundedLinearOperators ((G . f),F))) is Element of BoundedLinearOperators ((G . f),F)
K213((R_VectorSpace_of_LinearOperators ((G . f),F)),(BoundedLinearOperators ((G . f),F))) is Relation-like K7((BoundedLinearOperators ((G . f),F)),(BoundedLinearOperators ((G . f),F))) -defined BoundedLinearOperators ((G . f),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . f),F)),(BoundedLinearOperators ((G . f),F))),(BoundedLinearOperators ((G . f),F))))
K7((BoundedLinearOperators ((G . f),F)),(BoundedLinearOperators ((G . f),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . f),F)),(BoundedLinearOperators ((G . f),F))),(BoundedLinearOperators ((G . f),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . f),F)),(BoundedLinearOperators ((G . f),F))),(BoundedLinearOperators ((G . f),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . f),F)),(BoundedLinearOperators ((G . f),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . f),F))) -defined BoundedLinearOperators ((G . f),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . f),F))),(BoundedLinearOperators ((G . f),F))))
K7(REAL,(BoundedLinearOperators ((G . f),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . f),F))),(BoundedLinearOperators ((G . f),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . f),F))),(BoundedLinearOperators ((G . f),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . f),F) is non empty Relation-like BoundedLinearOperators ((G . f),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . f),F)),REAL))
K7((BoundedLinearOperators ((G . f),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . f),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . f),F)),K215((R_VectorSpace_of_LinearOperators ((G . f),F)),(BoundedLinearOperators ((G . f),F))),K213((R_VectorSpace_of_LinearOperators ((G . f),F)),(BoundedLinearOperators ((G . f),F))),K214((R_VectorSpace_of_LinearOperators ((G . f),F)),(BoundedLinearOperators ((G . f),F))),(BoundedLinearOperatorsNorm ((G . f),F)) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . f),F)) is non empty set
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of F is non empty non trivial set
K7( the carrier of (product G), the carrier of F) is Relation-like set
K6(K7( the carrier of (product G), the carrier of F)) is set
f is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of F))
dom f is Element of K6( the carrier of (product G))
K6( the carrier of (product G)) is set
X is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
m is set
(G,m) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . (G,m) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,m)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,m)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,m)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,m)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,m)),F))
the carrier of (G . (G,m)) is non empty non trivial set
K266( the carrier of (G . (G,m)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,m)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,m)), the carrier of F
FuncZero ( the carrier of (G . (G,m)),F) is Relation-like the carrier of (G . (G,m)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,m)), the carrier of F)
0. F is V68(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
the carrier of (G . (G,m)) --> (0. F) is non empty Relation-like the carrier of (G . (G,m)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,m)), the carrier of F))
K7( the carrier of (G . (G,m)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,m)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,m)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,m)), the carrier of F)),(Funcs ( the carrier of (G . (G,m)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,m)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,m)), the carrier of F)),(Funcs ( the carrier of (G . (G,m)), the carrier of F))),(Funcs ( the carrier of (G . (G,m)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,m)), the carrier of F)),(Funcs ( the carrier of (G . (G,m)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,m)), the carrier of F)),(Funcs ( the carrier of (G . (G,m)), the carrier of F))),(Funcs ( the carrier of (G . (G,m)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,m)), the carrier of F)),(Funcs ( the carrier of (G . (G,m)), the carrier of F))),(Funcs ( the carrier of (G . (G,m)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,m)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,m)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,m)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,m)), the carrier of F))),(Funcs ( the carrier of (G . (G,m)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,m)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,m)), the carrier of F))),(Funcs ( the carrier of (G . (G,m)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,m)), the carrier of F))),(Funcs ( the carrier of (G . (G,m)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,m)), the carrier of F)),(FuncZero ( the carrier of (G . (G,m)),F)),(FuncAdd ( the carrier of (G . (G,m)),F)),(FuncExtMult ( the carrier of (G . (G,m)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,m)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,m)),F)) is set
K215(K266( the carrier of (G . (G,m)),F),(LinearOperators ((G . (G,m)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,m)),F)
K213(K266( the carrier of (G . (G,m)),F),(LinearOperators ((G . (G,m)),F))) is Relation-like K7((LinearOperators ((G . (G,m)),F)),(LinearOperators ((G . (G,m)),F))) -defined LinearOperators ((G . (G,m)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,m)),F)),(LinearOperators ((G . (G,m)),F))),(LinearOperators ((G . (G,m)),F))))
K7((LinearOperators ((G . (G,m)),F)),(LinearOperators ((G . (G,m)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,m)),F)),(LinearOperators ((G . (G,m)),F))),(LinearOperators ((G . (G,m)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,m)),F)),(LinearOperators ((G . (G,m)),F))),(LinearOperators ((G . (G,m)),F)))) is set
K214(K266( the carrier of (G . (G,m)),F),(LinearOperators ((G . (G,m)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,m)),F))) -defined LinearOperators ((G . (G,m)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,m)),F))),(LinearOperators ((G . (G,m)),F))))
K7(REAL,(LinearOperators ((G . (G,m)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,m)),F))),(LinearOperators ((G . (G,m)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,m)),F))),(LinearOperators ((G . (G,m)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,m)),F)),K215(K266( the carrier of (G . (G,m)),F),(LinearOperators ((G . (G,m)),F))),K213(K266( the carrier of (G . (G,m)),F),(LinearOperators ((G . (G,m)),F))),K214(K266( the carrier of (G . (G,m)),F),(LinearOperators ((G . (G,m)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,m)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,m)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,m)),F)),(BoundedLinearOperators ((G . (G,m)),F))) is Element of BoundedLinearOperators ((G . (G,m)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,m)),F)),(BoundedLinearOperators ((G . (G,m)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,m)),F)),(BoundedLinearOperators ((G . (G,m)),F))) -defined BoundedLinearOperators ((G . (G,m)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,m)),F)),(BoundedLinearOperators ((G . (G,m)),F))),(BoundedLinearOperators ((G . (G,m)),F))))
K7((BoundedLinearOperators ((G . (G,m)),F)),(BoundedLinearOperators ((G . (G,m)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,m)),F)),(BoundedLinearOperators ((G . (G,m)),F))),(BoundedLinearOperators ((G . (G,m)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,m)),F)),(BoundedLinearOperators ((G . (G,m)),F))),(BoundedLinearOperators ((G . (G,m)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,m)),F)),(BoundedLinearOperators ((G . (G,m)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,m)),F))) -defined BoundedLinearOperators ((G . (G,m)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,m)),F))),(BoundedLinearOperators ((G . (G,m)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,m)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,m)),F))),(BoundedLinearOperators ((G . (G,m)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,m)),F))),(BoundedLinearOperators ((G . (G,m)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,m)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,m)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,m)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,m)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,m)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,m)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,m)),F)),(BoundedLinearOperators ((G . (G,m)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,m)),F)),(BoundedLinearOperators ((G . (G,m)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,m)),F)),(BoundedLinearOperators ((G . (G,m)),F))),(BoundedLinearOperatorsNorm ((G . (G,m)),F)) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F)) is non empty set
(G,(G,m),X) is non empty Relation-like the carrier of (G . (G,m)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,m)), the carrier of (product G)))
K7( the carrier of (G . (G,m)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,m)), the carrier of (product G))) is set
i is V11() real ext-real Element of REAL
i0 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F))
y0 is Element of the carrier of (G . (G,m))
r is Element of the carrier of (G . (G,m))
((G . (G,m)),y0,r) is Element of K6( the carrier of (G . (G,m)))
K6( the carrier of (G . (G,m))) is set
r - y0 is Element of the carrier of (G . (G,m))
- y0 is Element of the carrier of (G . (G,m))
r + (- y0) is Element of the carrier of (G . (G,m))
the addF of (G . (G,m)) is Relation-like K7( the carrier of (G . (G,m)), the carrier of (G . (G,m))) -defined the carrier of (G . (G,m)) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . (G,m)), the carrier of (G . (G,m))), the carrier of (G . (G,m))))
K7( the carrier of (G . (G,m)), the carrier of (G . (G,m))) is Relation-like set
K7(K7( the carrier of (G . (G,m)), the carrier of (G . (G,m))), the carrier of (G . (G,m))) is Relation-like set
K6(K7(K7( the carrier of (G . (G,m)), the carrier of (G . (G,m))), the carrier of (G . (G,m)))) is set
K462( the carrier of (G . (G,m)), the addF of (G . (G,m)),r,(- y0)) is Element of the carrier of (G . (G,m))
{ (y0 + (b₁ * (r - y0))) where b₁ is V11() real ext-real Element of REAL : ( not b₁ <= 0 & not 1 <= b₁ ) } is set
LSeg (y0,r) is Element of K6( the carrier of (G . (G,m)))
(G,(G,m),X) . r is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f /. ((G,(G,m),X) . r) is Element of the carrier of F
(G,(G,m),X) . y0 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f /. ((G,(G,m),X) . y0) is Element of the carrier of F
(f /. ((G,(G,m),X) . r)) - (f /. ((G,(G,m),X) . y0)) is Element of the carrier of F
- (f /. ((G,(G,m),X) . y0)) is Element of the carrier of F
(f /. ((G,(G,m),X) . r)) + (- (f /. ((G,(G,m),X) . y0))) is Element of the carrier of F
the addF of F is Relation-like K7( the carrier of F, the carrier of F) -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of F, the carrier of F), the carrier of F))
K7( the carrier of F, the carrier of F) is Relation-like set
K7(K7( the carrier of F, the carrier of F), the carrier of F) is Relation-like set
K6(K7(K7( the carrier of F, the carrier of F), the carrier of F)) is set
K462( the carrier of F, the addF of F,(f /. ((G,(G,m),X) . r)),(- (f /. ((G,(G,m),X) . y0)))) is Element of the carrier of F
i0 . (r - y0) is Element of the carrier of F
((f /. ((G,(G,m),X) . r)) - (f /. ((G,(G,m),X) . y0))) - (i0 . (r - y0)) is Element of the carrier of F
- (i0 . (r - y0)) is Element of the carrier of F
((f /. ((G,(G,m),X) . r)) - (f /. ((G,(G,m),X) . y0))) + (- (i0 . (r - y0))) is Element of the carrier of F
K462( the carrier of F, the addF of F,((f /. ((G,(G,m),X) . r)) - (f /. ((G,(G,m),X) . y0))),(- (i0 . (r - y0)))) is Element of the carrier of F
||.(((f /. ((G,(G,m),X) . r)) - (f /. ((G,(G,m),X) . y0))) - (i0 . (r - y0))).|| is V11() real ext-real non negative Element of REAL
the normF of F is non empty Relation-like the carrier of F -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of F,REAL))
K7( the carrier of F,REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of F,REAL)) is non trivial V49() set
the normF of F . (((f /. ((G,(G,m),X) . r)) - (f /. ((G,(G,m),X) . y0))) - (i0 . (r - y0))) is V11() real ext-real Element of REAL
||.(r - y0).|| is V11() real ext-real non negative Element of REAL
the normF of (G . (G,m)) is non empty Relation-like the carrier of (G . (G,m)) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (G . (G,m)),REAL))
K7( the carrier of (G . (G,m)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (G . (G,m)),REAL)) is non trivial V49() set
the normF of (G . (G,m)) . (r - y0) is V11() real ext-real Element of REAL
i * ||.(r - y0).|| is V11() real ext-real Element of REAL
f * (G,(G,m),X) is Relation-like the carrier of (G . (G,m)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,m)), the carrier of F))
DD is set
dom (G,(G,m),X) is non empty Element of K6( the carrier of (G . (G,m)))
a is Element of the carrier of (G . (G,m))
(G,(G,m),X) . a is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
dom (f * (G,(G,m),X)) is Element of K6( the carrier of (G . (G,m)))
DD is Element of the carrier of (G . (G,m))
(G,(G,m),X) . DD is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,(G,m)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,m)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,m))))
K7( the carrier of (product G), the carrier of (G . (G,m))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,m)))) is set
(G,(G,m)) . ((G,(G,m),X) . DD) is Element of the carrier of (G . (G,m))
(G,(G,m),((G,(G,m),X) . DD)) is non empty Relation-like the carrier of (G . (G,m)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,m)), the carrier of (product G)))
f * (G,(G,m),((G,(G,m),X) . DD)) is Relation-like the carrier of (G . (G,m)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,m)), the carrier of F))
DD is set
a is V11() real ext-real Element of REAL
a * (r - y0) is Element of the carrier of (G . (G,m))
the Mult of (G . (G,m)) is Relation-like K7(REAL, the carrier of (G . (G,m))) -defined the carrier of (G . (G,m)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (G . (G,m))), the carrier of (G . (G,m))))
K7(REAL, the carrier of (G . (G,m))) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of (G . (G,m))), the carrier of (G . (G,m))) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of (G . (G,m))), the carrier of (G . (G,m)))) is non trivial V49() set
K458( the Mult of (G . (G,m)),a,(r - y0)) is set
y0 + (a * (r - y0)) is Element of the carrier of (G . (G,m))
K462( the carrier of (G . (G,m)), the addF of (G . (G,m)),y0,(a * (r - y0))) is Element of the carrier of (G . (G,m))
1 - a is V11() real ext-real Element of REAL
K38(a) is V11() real ext-real set
K36(1,K38(a)) is V11() real ext-real set
(1 - a) * y0 is Element of the carrier of (G . (G,m))
K458( the Mult of (G . (G,m)),(1 - a),y0) is set
a * r is Element of the carrier of (G . (G,m))
K458( the Mult of (G . (G,m)),a,r) is set
((1 - a) * y0) + (a * r) is Element of the carrier of (G . (G,m))
K462( the carrier of (G . (G,m)), the addF of (G . (G,m)),((1 - a) * y0),(a * r)) is Element of the carrier of (G . (G,m))
{ (((1 - b₁) * y0) + (b₁ * r)) where b₁ is V11() real ext-real Element of REAL : ( 0 <= b₁ & b₁ <= 1 ) } is set
DD is Element of the carrier of (G . (G,m))
DD is Element of the carrier of (G . (G,m))
DD is Element of the carrier of (G . (G,m))
(G,(G,m),X) . DD is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,F,m,f,((G,(G,m),X) . DD)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F))
(G,(G,m),((G,(G,m),X) . DD)) is non empty Relation-like the carrier of (G . (G,m)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,m)), the carrier of (product G)))
f * (G,(G,m),((G,(G,m),X) . DD)) is Relation-like the carrier of (G . (G,m)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,m)), the carrier of F))
(G,(G,m)) . ((G,(G,m),X) . DD) is Element of the carrier of (G . (G,m))
diff ((f * (G,(G,m),((G,(G,m),X) . DD))),((G,(G,m)) . ((G,(G,m),X) . DD))) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F))
(G,F,m,f,((G,(G,m),X) . DD)) - i0 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F))
- i0 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F))
(G,F,m,f,((G,(G,m),X) . DD)) + (- i0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F))
the addF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F)) is Relation-like K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F))) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F)) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F))))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F))) is Relation-like set
K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F))) is Relation-like set
K6(K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F)))) is set
K462( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F)), the addF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F)),(G,F,m,f,((G,(G,m),X) . DD)),(- i0)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F))
||.((G,F,m,f,((G,(G,m),X) . DD)) - i0).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F)) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F)) -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 ((G . (G,m)),F)),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F)),REAL)) is non trivial V49() set
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F)) . ((G,F,m,f,((G,(G,m),X) . DD)) - i0) is V11() real ext-real Element of REAL
diff ((f * (G,(G,m),X)),DD) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F))
(diff ((f * (G,(G,m),X)),DD)) - i0 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F))
(diff ((f * (G,(G,m),X)),DD)) + (- i0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F))
K462( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F)), the addF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F)),(diff ((f * (G,(G,m),X)),DD)),(- i0)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F))
||.((diff ((f * (G,(G,m),X)),DD)) - i0).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,m)),F)) . ((diff ((f * (G,(G,m),X)),DD)) - i0) is V11() real ext-real Element of REAL
(G,(G,m),X) /. y0 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f /. ((G,(G,m),X) /. y0) is Element of the carrier of F
(G,(G,m),X) /. r is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f /. ((G,(G,m),X) /. r) is Element of the carrier of F
(f * (G,(G,m),X)) /. y0 is Element of the carrier of F
(f * (G,(G,m),X)) /. r is Element of the carrier of F
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
f is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
F is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f - F is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
- F is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f + (- F) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
the addF of (product G) is Relation-like K7( the carrier of (product G), the carrier of (product G)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)))
K7( the carrier of (product G), the carrier of (product G)) is Relation-like set
K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)) is Relation-like set
K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G))) is set
K462( the carrier of (product G), the addF of (product G),f,(- F)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.(f - F).|| is V11() real ext-real non negative Element of REAL
the normF of (product G) is non empty Relation-like the carrier of (product G) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (product G),REAL))
K7( the carrier of (product G),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (product G),REAL)) is non trivial V49() set
the normF of (product G) . (f - F) is V11() real ext-real Element of REAL
X is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
X - F is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
X + (- F) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),X,(- F)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.(X - F).|| is V11() real ext-real non negative Element of REAL
the normF of (product G) . (X - F) is V11() real ext-real Element of REAL
m is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
m - F is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
m + (- F) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),m,(- F)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.(m - F).|| is V11() real ext-real non negative Element of REAL
the normF of (product G) . (m - F) is V11() real ext-real Element of REAL
i is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . i is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . i) is non empty non trivial set
(G,i) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . i) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . i)))
K7( the carrier of (product G), the carrier of (G . i)) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . i))) is set
(G,i) . f is Element of the carrier of (G . i)
(G,i,f) is non empty Relation-like the carrier of (G . i) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . i), the carrier of (product G)))
K7( the carrier of (G . i), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . i), the carrier of (product G))) is set
i0 is V11() real ext-real Element of REAL
y0 is Element of the carrier of (G . i)
r is Element of the carrier of (G . i)
(G,i,f) . r is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
s is Element of the carrier of (G . i)
LSeg (y0,r) is Element of K6( the carrier of (G . i))
K6( the carrier of (G . i)) is set
(G,i,f) . s is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,i) . F is Element of the carrier of (G . i)
{ (((1 - b₁) * y0) + (b₁ * r)) where b₁ is V11() real ext-real Element of REAL : ( 0 <= b₁ & b₁ <= 1 ) } is set
t is V11() real ext-real Element of REAL
1 - t is V11() real ext-real Element of REAL
K38(t) is V11() real ext-real set
K36(1,K38(t)) is V11() real ext-real set
(1 - t) * y0 is Element of the carrier of (G . i)
the Mult of (G . i) is Relation-like K7(REAL, the carrier of (G . i)) -defined the carrier of (G . i) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (G . i)), the carrier of (G . i)))
K7(REAL, the carrier of (G . i)) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of (G . i)), the carrier of (G . i)) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of (G . i)), the carrier of (G . i))) is non trivial V49() set
K458( the Mult of (G . i),(1 - t),y0) is set
t * r is Element of the carrier of (G . i)
K458( the Mult of (G . i),t,r) is set
((1 - t) * y0) + (t * r) is Element of the carrier of (G . i)
the addF of (G . i) is Relation-like K7( the carrier of (G . i), the carrier of (G . i)) -defined the carrier of (G . i) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . i), the carrier of (G . i)), the carrier of (G . i)))
K7( the carrier of (G . i), the carrier of (G . i)) is Relation-like set
K7(K7( the carrier of (G . i), the carrier of (G . i)), the carrier of (G . i)) is Relation-like set
K6(K7(K7( the carrier of (G . i), the carrier of (G . i)), the carrier of (G . i))) is set
K462( the carrier of (G . i), the addF of (G . i),((1 - t) * y0),(t * r)) is Element of the carrier of (G . i)
r - y0 is Element of the carrier of (G . i)
- y0 is Element of the carrier of (G . i)
r + (- y0) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),r,(- y0)) is Element of the carrier of (G . i)
t * (r - y0) is Element of the carrier of (G . i)
K458( the Mult of (G . i),t,(r - y0)) is set
y0 + (t * (r - y0)) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),y0,(t * (r - y0))) is Element of the carrier of (G . i)
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
tm is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
normsequence (G,tm) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() len G -element FinSequence-like FinSubsequence-like countable Element of REAL (len G)
REAL (len G) is non empty functional FinSequence-membered FinSequenceSet of REAL
(len G) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
REAL * is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like V49() FinSequence-like FinSubsequence-like countable Element of REAL * : len b₁ = len G } is set
Sw1 is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
normsequence (G,Sw1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() len G -element FinSequence-like FinSubsequence-like countable Element of REAL (len G)
w2 is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
normsequence (G,w2) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() len G -element FinSequence-like FinSubsequence-like countable Element of REAL (len G)
(1 - t) * (f - F) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
the Mult of (product G) is Relation-like K7(REAL, the carrier of (product G)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (product G)), the carrier of (product G)))
K7(REAL, the carrier of (product G)) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of (product G)), the carrier of (product G)) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of (product G)), the carrier of (product G))) is non trivial V49() set
K458( the Mult of (product G),(1 - t),(f - F)) is set
t * (X - F) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K458( the Mult of (product G),t,(X - F)) is set
((1 - t) * (f - F)) + (t * (X - F)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),((1 - t) * (f - F)),(t * (X - F))) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
p1 is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
normsequence (G,p1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() len G -element FinSequence-like FinSubsequence-like countable Element of REAL (len G)
(1 - t) + t is V11() real ext-real Element of REAL
t * y0 is Element of the carrier of (G . i)
K458( the Mult of (G . i),t,y0) is set
(t * r) - (t * y0) is Element of the carrier of (G . i)
- (t * y0) is Element of the carrier of (G . i)
(t * r) + (- (t * y0)) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),(t * r),(- (t * y0))) is Element of the carrier of (G . i)
y0 + ((t * r) - (t * y0)) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),y0,((t * r) - (t * y0))) is Element of the carrier of (G . i)
y0 + (- (t * y0)) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),y0,(- (t * y0))) is Element of the carrier of (G . i)
(y0 + (- (t * y0))) + (t * r) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),(y0 + (- (t * y0))),(t * r)) is Element of the carrier of (G . i)
1 * y0 is Element of the carrier of (G . i)
K458( the Mult of (G . i),1,y0) is set
(1 * y0) - (t * y0) is Element of the carrier of (G . i)
(1 * y0) + (- (t * y0)) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),(1 * y0),(- (t * y0))) is Element of the carrier of (G . i)
((1 * y0) - (t * y0)) + (t * r) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),((1 * y0) - (t * y0)),(t * r)) is Element of the carrier of (G . i)
a is Element of the carrier of (G . i)
s - a is Element of the carrier of (G . i)
- a is Element of the carrier of (G . i)
s + (- a) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),s,(- a)) is Element of the carrier of (G . i)
1 * a is Element of the carrier of (G . i)
K458( the Mult of (G . i),1,a) is set
(((1 - t) * y0) + (t * r)) - (1 * a) is Element of the carrier of (G . i)
- (1 * a) is Element of the carrier of (G . i)
(((1 - t) * y0) + (t * r)) + (- (1 * a)) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),(((1 - t) * y0) + (t * r)),(- (1 * a))) is Element of the carrier of (G . i)
(1 - t) * a is Element of the carrier of (G . i)
K458( the Mult of (G . i),(1 - t),a) is set
t * a is Element of the carrier of (G . i)
K458( the Mult of (G . i),t,a) is set
((1 - t) * a) + (t * a) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),((1 - t) * a),(t * a)) is Element of the carrier of (G . i)
(((1 - t) * y0) + (t * r)) - (((1 - t) * a) + (t * a)) is Element of the carrier of (G . i)
- (((1 - t) * a) + (t * a)) is Element of the carrier of (G . i)
(((1 - t) * y0) + (t * r)) + (- (((1 - t) * a) + (t * a))) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),(((1 - t) * y0) + (t * r)),(- (((1 - t) * a) + (t * a)))) is Element of the carrier of (G . i)
(((1 - t) * y0) + (t * r)) - (t * a) is Element of the carrier of (G . i)
- (t * a) is Element of the carrier of (G . i)
(((1 - t) * y0) + (t * r)) + (- (t * a)) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),(((1 - t) * y0) + (t * r)),(- (t * a))) is Element of the carrier of (G . i)
((((1 - t) * y0) + (t * r)) - (t * a)) - ((1 - t) * a) is Element of the carrier of (G . i)
- ((1 - t) * a) is Element of the carrier of (G . i)
((((1 - t) * y0) + (t * r)) - (t * a)) + (- ((1 - t) * a)) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),((((1 - t) * y0) + (t * r)) - (t * a)),(- ((1 - t) * a))) is Element of the carrier of (G . i)
(t * r) - (t * a) is Element of the carrier of (G . i)
(t * r) + (- (t * a)) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),(t * r),(- (t * a))) is Element of the carrier of (G . i)
((1 - t) * y0) + ((t * r) - (t * a)) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),((1 - t) * y0),((t * r) - (t * a))) is Element of the carrier of (G . i)
(((1 - t) * y0) + ((t * r) - (t * a))) - ((1 - t) * a) is Element of the carrier of (G . i)
(((1 - t) * y0) + ((t * r) - (t * a))) + (- ((1 - t) * a)) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),(((1 - t) * y0) + ((t * r) - (t * a))),(- ((1 - t) * a))) is Element of the carrier of (G . i)
((1 - t) * y0) - ((1 - t) * a) is Element of the carrier of (G . i)
((1 - t) * y0) + (- ((1 - t) * a)) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),((1 - t) * y0),(- ((1 - t) * a))) is Element of the carrier of (G . i)
((t * r) - (t * a)) + (((1 - t) * y0) - ((1 - t) * a)) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),((t * r) - (t * a)),(((1 - t) * y0) - ((1 - t) * a))) is Element of the carrier of (G . i)
r - a is Element of the carrier of (G . i)
r + (- a) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),r,(- a)) is Element of the carrier of (G . i)
t * (r - a) is Element of the carrier of (G . i)
K458( the Mult of (G . i),t,(r - a)) is set
(t * (r - a)) + (((1 - t) * y0) - ((1 - t) * a)) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),(t * (r - a)),(((1 - t) * y0) - ((1 - t) * a))) is Element of the carrier of (G . i)
y0 - a is Element of the carrier of (G . i)
y0 + (- a) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),y0,(- a)) is Element of the carrier of (G . i)
(1 - t) * (y0 - a) is Element of the carrier of (G . i)
K458( the Mult of (G . i),(1 - t),(y0 - a)) is set
(t * (r - a)) + ((1 - t) * (y0 - a)) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),(t * (r - a)),((1 - t) * (y0 - a))) is Element of the carrier of (G . i)
P1 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable set
dom P1 is V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
Seg (len G) is non empty V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= len G ) } is set
P0 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable set
dom P0 is V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
PP is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
P1 . PP is set
P0 . PP is set
(G,i) . tm is set
(G,i) . m is Element of the carrier of (G . i)
((G,i) . m) - ((G,i) . F) is Element of the carrier of (G . i)
- ((G,i) . F) is Element of the carrier of (G . i)
((G,i) . m) + (- ((G,i) . F)) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),((G,i) . m),(- ((G,i) . F))) is Element of the carrier of (G . i)
(G,i) . X is Element of the carrier of (G . i)
(G,i) . p1 is set
(G,i) . ((1 - t) * (f - F)) is Element of the carrier of (G . i)
(G,i) . (t * (X - F)) is Element of the carrier of (G . i)
((G,i) . ((1 - t) * (f - F))) + ((G,i) . (t * (X - F))) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),((G,i) . ((1 - t) * (f - F))),((G,i) . (t * (X - F)))) is Element of the carrier of (G . i)
(G,i) . (f - F) is Element of the carrier of (G . i)
(1 - t) * ((G,i) . (f - F)) is Element of the carrier of (G . i)
K458( the Mult of (G . i),(1 - t),((G,i) . (f - F))) is set
((1 - t) * ((G,i) . (f - F))) + ((G,i) . (t * (X - F))) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),((1 - t) * ((G,i) . (f - F))),((G,i) . (t * (X - F)))) is Element of the carrier of (G . i)
(G,i) . (X - F) is Element of the carrier of (G . i)
t * ((G,i) . (X - F)) is Element of the carrier of (G . i)
K458( the Mult of (G . i),t,((G,i) . (X - F))) is set
((1 - t) * ((G,i) . (f - F))) + (t * ((G,i) . (X - F))) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),((1 - t) * ((G,i) . (f - F))),(t * ((G,i) . (X - F)))) is Element of the carrier of (G . i)
((G,i) . f) - ((G,i) . F) is Element of the carrier of (G . i)
((G,i) . f) + (- ((G,i) . F)) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),((G,i) . f),(- ((G,i) . F))) is Element of the carrier of (G . i)
(1 - t) * (((G,i) . f) - ((G,i) . F)) is Element of the carrier of (G . i)
K458( the Mult of (G . i),(1 - t),(((G,i) . f) - ((G,i) . F))) is set
((1 - t) * (((G,i) . f) - ((G,i) . F))) + (t * ((G,i) . (X - F))) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),((1 - t) * (((G,i) . f) - ((G,i) . F))),(t * ((G,i) . (X - F)))) is Element of the carrier of (G . i)
((1 - t) * (y0 - a)) + (t * (r - a)) is Element of the carrier of (G . i)
K462( the carrier of (G . i), the addF of (G . i),((1 - t) * (y0 - a)),(t * (r - a))) is Element of the carrier of (G . i)
pt is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . pt is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . pt) is non empty non trivial set
(G,pt) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . pt) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . pt)))
K7( the carrier of (product G), the carrier of (G . pt)) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . pt))) is set
(G,pt) . f is Element of the carrier of (G . pt)
(G,pt) . m is Element of the carrier of (G . pt)
(G,pt) . X is Element of the carrier of (G . pt)
(G,pt) . tm is set
(G,pt) . F is Element of the carrier of (G . pt)
((G,pt) . m) - ((G,pt) . F) is Element of the carrier of (G . pt)
- ((G,pt) . F) is Element of the carrier of (G . pt)
((G,pt) . m) + (- ((G,pt) . F)) is Element of the carrier of (G . pt)
the addF of (G . pt) is Relation-like K7( the carrier of (G . pt), the carrier of (G . pt)) -defined the carrier of (G . pt) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . pt), the carrier of (G . pt)), the carrier of (G . pt)))
K7( the carrier of (G . pt), the carrier of (G . pt)) is Relation-like set
K7(K7( the carrier of (G . pt), the carrier of (G . pt)), the carrier of (G . pt)) is Relation-like set
K6(K7(K7( the carrier of (G . pt), the carrier of (G . pt)), the carrier of (G . pt))) is set
K462( the carrier of (G . pt), the addF of (G . pt),((G,pt) . m),(- ((G,pt) . F))) is Element of the carrier of (G . pt)
(G,pt) . p1 is set
(G,pt) . ((1 - t) * (f - F)) is Element of the carrier of (G . pt)
(G,pt) . (t * (X - F)) is Element of the carrier of (G . pt)
((G,pt) . ((1 - t) * (f - F))) + ((G,pt) . (t * (X - F))) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),((G,pt) . ((1 - t) * (f - F))),((G,pt) . (t * (X - F)))) is Element of the carrier of (G . pt)
(G,pt) . (f - F) is Element of the carrier of (G . pt)
(1 - t) * ((G,pt) . (f - F)) is Element of the carrier of (G . pt)
the Mult of (G . pt) is Relation-like K7(REAL, the carrier of (G . pt)) -defined the carrier of (G . pt) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (G . pt)), the carrier of (G . pt)))
K7(REAL, the carrier of (G . pt)) is non trivial Relation-like V49() set
K7(K7(REAL, the carrier of (G . pt)), the carrier of (G . pt)) is non trivial Relation-like V49() set
K6(K7(K7(REAL, the carrier of (G . pt)), the carrier of (G . pt))) is non trivial V49() set
K458( the Mult of (G . pt),(1 - t),((G,pt) . (f - F))) is set
((1 - t) * ((G,pt) . (f - F))) + ((G,pt) . (t * (X - F))) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),((1 - t) * ((G,pt) . (f - F))),((G,pt) . (t * (X - F)))) is Element of the carrier of (G . pt)
(G,pt) . (X - F) is Element of the carrier of (G . pt)
t * ((G,pt) . (X - F)) is Element of the carrier of (G . pt)
K458( the Mult of (G . pt),t,((G,pt) . (X - F))) is set
((1 - t) * ((G,pt) . (f - F))) + (t * ((G,pt) . (X - F))) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),((1 - t) * ((G,pt) . (f - F))),(t * ((G,pt) . (X - F)))) is Element of the carrier of (G . pt)
((G,pt) . f) - ((G,pt) . F) is Element of the carrier of (G . pt)
((G,pt) . f) + (- ((G,pt) . F)) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),((G,pt) . f),(- ((G,pt) . F))) is Element of the carrier of (G . pt)
(1 - t) * (((G,pt) . f) - ((G,pt) . F)) is Element of the carrier of (G . pt)
K458( the Mult of (G . pt),(1 - t),(((G,pt) . f) - ((G,pt) . F))) is set
((1 - t) * (((G,pt) . f) - ((G,pt) . F))) + (t * ((G,pt) . (X - F))) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),((1 - t) * (((G,pt) . f) - ((G,pt) . F))),(t * ((G,pt) . (X - F)))) is Element of the carrier of (G . pt)
t * (((G,pt) . f) - ((G,pt) . F)) is Element of the carrier of (G . pt)
K458( the Mult of (G . pt),t,(((G,pt) . f) - ((G,pt) . F))) is set
((1 - t) * (((G,pt) . f) - ((G,pt) . F))) + (t * (((G,pt) . f) - ((G,pt) . F))) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),((1 - t) * (((G,pt) . f) - ((G,pt) . F))),(t * (((G,pt) . f) - ((G,pt) . F)))) is Element of the carrier of (G . pt)
(1 - t) * ((G,pt) . f) is Element of the carrier of (G . pt)
K458( the Mult of (G . pt),(1 - t),((G,pt) . f)) is set
(1 - t) * ((G,pt) . F) is Element of the carrier of (G . pt)
K458( the Mult of (G . pt),(1 - t),((G,pt) . F)) is set
((1 - t) * ((G,pt) . f)) - ((1 - t) * ((G,pt) . F)) is Element of the carrier of (G . pt)
- ((1 - t) * ((G,pt) . F)) is Element of the carrier of (G . pt)
((1 - t) * ((G,pt) . f)) + (- ((1 - t) * ((G,pt) . F))) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),((1 - t) * ((G,pt) . f)),(- ((1 - t) * ((G,pt) . F)))) is Element of the carrier of (G . pt)
(((1 - t) * ((G,pt) . f)) - ((1 - t) * ((G,pt) . F))) + (t * (((G,pt) . f) - ((G,pt) . F))) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),(((1 - t) * ((G,pt) . f)) - ((1 - t) * ((G,pt) . F))),(t * (((G,pt) . f) - ((G,pt) . F)))) is Element of the carrier of (G . pt)
t * ((G,pt) . f) is Element of the carrier of (G . pt)
K458( the Mult of (G . pt),t,((G,pt) . f)) is set
t * ((G,pt) . F) is Element of the carrier of (G . pt)
K458( the Mult of (G . pt),t,((G,pt) . F)) is set
(t * ((G,pt) . f)) - (t * ((G,pt) . F)) is Element of the carrier of (G . pt)
- (t * ((G,pt) . F)) is Element of the carrier of (G . pt)
(t * ((G,pt) . f)) + (- (t * ((G,pt) . F))) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),(t * ((G,pt) . f)),(- (t * ((G,pt) . F)))) is Element of the carrier of (G . pt)
(((1 - t) * ((G,pt) . f)) - ((1 - t) * ((G,pt) . F))) + ((t * ((G,pt) . f)) - (t * ((G,pt) . F))) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),(((1 - t) * ((G,pt) . f)) - ((1 - t) * ((G,pt) . F))),((t * ((G,pt) . f)) - (t * ((G,pt) . F)))) is Element of the carrier of (G . pt)
(((1 - t) * ((G,pt) . f)) - ((1 - t) * ((G,pt) . F))) + (t * ((G,pt) . f)) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),(((1 - t) * ((G,pt) . f)) - ((1 - t) * ((G,pt) . F))),(t * ((G,pt) . f))) is Element of the carrier of (G . pt)
((((1 - t) * ((G,pt) . f)) - ((1 - t) * ((G,pt) . F))) + (t * ((G,pt) . f))) - (t * ((G,pt) . F)) is Element of the carrier of (G . pt)
((((1 - t) * ((G,pt) . f)) - ((1 - t) * ((G,pt) . F))) + (t * ((G,pt) . f))) + (- (t * ((G,pt) . F))) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),((((1 - t) * ((G,pt) . f)) - ((1 - t) * ((G,pt) . F))) + (t * ((G,pt) . f))),(- (t * ((G,pt) . F)))) is Element of the carrier of (G . pt)
((1 - t) * ((G,pt) . F)) - (t * ((G,pt) . f)) is Element of the carrier of (G . pt)
- (t * ((G,pt) . f)) is Element of the carrier of (G . pt)
((1 - t) * ((G,pt) . F)) + (- (t * ((G,pt) . f))) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),((1 - t) * ((G,pt) . F)),(- (t * ((G,pt) . f)))) is Element of the carrier of (G . pt)
((1 - t) * ((G,pt) . f)) - (((1 - t) * ((G,pt) . F)) - (t * ((G,pt) . f))) is Element of the carrier of (G . pt)
- (((1 - t) * ((G,pt) . F)) - (t * ((G,pt) . f))) is Element of the carrier of (G . pt)
((1 - t) * ((G,pt) . f)) + (- (((1 - t) * ((G,pt) . F)) - (t * ((G,pt) . f)))) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),((1 - t) * ((G,pt) . f)),(- (((1 - t) * ((G,pt) . F)) - (t * ((G,pt) . f))))) is Element of the carrier of (G . pt)
(((1 - t) * ((G,pt) . f)) - (((1 - t) * ((G,pt) . F)) - (t * ((G,pt) . f)))) - (t * ((G,pt) . F)) is Element of the carrier of (G . pt)
(((1 - t) * ((G,pt) . f)) - (((1 - t) * ((G,pt) . F)) - (t * ((G,pt) . f)))) + (- (t * ((G,pt) . F))) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),(((1 - t) * ((G,pt) . f)) - (((1 - t) * ((G,pt) . F)) - (t * ((G,pt) . f)))),(- (t * ((G,pt) . F)))) is Element of the carrier of (G . pt)
(t * ((G,pt) . f)) + (- ((1 - t) * ((G,pt) . F))) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),(t * ((G,pt) . f)),(- ((1 - t) * ((G,pt) . F)))) is Element of the carrier of (G . pt)
((1 - t) * ((G,pt) . f)) + ((t * ((G,pt) . f)) + (- ((1 - t) * ((G,pt) . F)))) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),((1 - t) * ((G,pt) . f)),((t * ((G,pt) . f)) + (- ((1 - t) * ((G,pt) . F))))) is Element of the carrier of (G . pt)
(((1 - t) * ((G,pt) . f)) + ((t * ((G,pt) . f)) + (- ((1 - t) * ((G,pt) . F))))) - (t * ((G,pt) . F)) is Element of the carrier of (G . pt)
(((1 - t) * ((G,pt) . f)) + ((t * ((G,pt) . f)) + (- ((1 - t) * ((G,pt) . F))))) + (- (t * ((G,pt) . F))) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),(((1 - t) * ((G,pt) . f)) + ((t * ((G,pt) . f)) + (- ((1 - t) * ((G,pt) . F))))),(- (t * ((G,pt) . F)))) is Element of the carrier of (G . pt)
((1 - t) * ((G,pt) . f)) + (t * ((G,pt) . f)) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),((1 - t) * ((G,pt) . f)),(t * ((G,pt) . f))) is Element of the carrier of (G . pt)
(((1 - t) * ((G,pt) . f)) + (t * ((G,pt) . f))) + (- ((1 - t) * ((G,pt) . F))) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),(((1 - t) * ((G,pt) . f)) + (t * ((G,pt) . f))),(- ((1 - t) * ((G,pt) . F)))) is Element of the carrier of (G . pt)
((((1 - t) * ((G,pt) . f)) + (t * ((G,pt) . f))) + (- ((1 - t) * ((G,pt) . F)))) - (t * ((G,pt) . F)) is Element of the carrier of (G . pt)
((((1 - t) * ((G,pt) . f)) + (t * ((G,pt) . f))) + (- ((1 - t) * ((G,pt) . F)))) + (- (t * ((G,pt) . F))) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),((((1 - t) * ((G,pt) . f)) + (t * ((G,pt) . f))) + (- ((1 - t) * ((G,pt) . F)))),(- (t * ((G,pt) . F)))) is Element of the carrier of (G . pt)
((1 - t) + t) * ((G,pt) . f) is Element of the carrier of (G . pt)
K458( the Mult of (G . pt),((1 - t) + t),((G,pt) . f)) is set
(((1 - t) + t) * ((G,pt) . f)) + (- ((1 - t) * ((G,pt) . F))) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),(((1 - t) + t) * ((G,pt) . f)),(- ((1 - t) * ((G,pt) . F)))) is Element of the carrier of (G . pt)
((((1 - t) + t) * ((G,pt) . f)) + (- ((1 - t) * ((G,pt) . F)))) - (t * ((G,pt) . F)) is Element of the carrier of (G . pt)
((((1 - t) + t) * ((G,pt) . f)) + (- ((1 - t) * ((G,pt) . F)))) + (- (t * ((G,pt) . F))) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),((((1 - t) + t) * ((G,pt) . f)) + (- ((1 - t) * ((G,pt) . F)))),(- (t * ((G,pt) . F)))) is Element of the carrier of (G . pt)
((G,pt) . f) + (- ((1 - t) * ((G,pt) . F))) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),((G,pt) . f),(- ((1 - t) * ((G,pt) . F)))) is Element of the carrier of (G . pt)
(((G,pt) . f) + (- ((1 - t) * ((G,pt) . F)))) - (t * ((G,pt) . F)) is Element of the carrier of (G . pt)
(((G,pt) . f) + (- ((1 - t) * ((G,pt) . F)))) + (- (t * ((G,pt) . F))) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),(((G,pt) . f) + (- ((1 - t) * ((G,pt) . F)))),(- (t * ((G,pt) . F)))) is Element of the carrier of (G . pt)
(- ((1 - t) * ((G,pt) . F))) - (t * ((G,pt) . F)) is Element of the carrier of (G . pt)
(- ((1 - t) * ((G,pt) . F))) + (- (t * ((G,pt) . F))) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),(- ((1 - t) * ((G,pt) . F))),(- (t * ((G,pt) . F)))) is Element of the carrier of (G . pt)
((G,pt) . f) + ((- ((1 - t) * ((G,pt) . F))) - (t * ((G,pt) . F))) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),((G,pt) . f),((- ((1 - t) * ((G,pt) . F))) - (t * ((G,pt) . F)))) is Element of the carrier of (G . pt)
(t * ((G,pt) . F)) + ((1 - t) * ((G,pt) . F)) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),(t * ((G,pt) . F)),((1 - t) * ((G,pt) . F))) is Element of the carrier of (G . pt)
- ((t * ((G,pt) . F)) + ((1 - t) * ((G,pt) . F))) is Element of the carrier of (G . pt)
((G,pt) . f) + (- ((t * ((G,pt) . F)) + ((1 - t) * ((G,pt) . F)))) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),((G,pt) . f),(- ((t * ((G,pt) . F)) + ((1 - t) * ((G,pt) . F))))) is Element of the carrier of (G . pt)
t + (1 - t) is V11() real ext-real Element of REAL
(t + (1 - t)) * ((G,pt) . F) is Element of the carrier of (G . pt)
K458( the Mult of (G . pt),(t + (1 - t)),((G,pt) . F)) is set
- ((t + (1 - t)) * ((G,pt) . F)) is Element of the carrier of (G . pt)
((G,pt) . f) + (- ((t + (1 - t)) * ((G,pt) . F))) is Element of the carrier of (G . pt)
K462( the carrier of (G . pt), the addF of (G . pt),((G,pt) . f),(- ((t + (1 - t)) * ((G,pt) . F)))) is Element of the carrier of (G . pt)
ys is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() len G -element FinSequence-like FinSubsequence-like countable FinSequence of REAL
len ys is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
p0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() len G -element FinSequence-like FinSubsequence-like countable FinSequence of REAL
len p0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
Seg (len ys) is V49() len ys -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= len ys ) } is set
PP is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
ys . PP is V11() real ext-real Element of REAL
p0 . PP is V11() real ext-real Element of REAL
pt is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . pt is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . pt) is non empty non trivial set
(G,tm,pt) is Element of the carrier of (G . pt)
G . pt is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (G . pt) is non empty set
g is Element of the carrier of (G . pt)
||.g.|| is V11() real ext-real non negative Element of REAL
the normF of (G . pt) is non empty Relation-like the carrier of (G . pt) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (G . pt),REAL))
K7( the carrier of (G . pt),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (G . pt),REAL)) is non trivial V49() set
the normF of (G . pt) . g is V11() real ext-real Element of REAL
P0 . PP is set
|.ys.| is V11() real ext-real non negative Element of REAL
sqr ys is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
Sum (sqr ys) is V11() real ext-real Element of REAL
sqrt (Sum (sqr ys)) is V11() real ext-real Element of REAL
|.p0.| is V11() real ext-real non negative Element of REAL
sqr p0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
Sum (sqr p0) is V11() real ext-real Element of REAL
sqrt (Sum (sqr p0)) is V11() real ext-real Element of REAL
productnorm G is non empty Relation-like product (carr G) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((product (carr G)),REAL))
K7((product (carr G)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((product (carr G)),REAL)) is non trivial V49() set
(productnorm G) . (m - F) is V11() real ext-real Element of REAL
||.(((1 - t) * (f - F)) + (t * (X - F))).|| is V11() real ext-real non negative Element of REAL
the normF of (product G) . (((1 - t) * (f - F)) + (t * (X - F))) is V11() real ext-real Element of REAL
(productnorm G) . (((1 - t) * (f - F)) + (t * (X - F))) is V11() real ext-real Element of REAL
|.(normsequence (G,p1)).| is V11() real ext-real non negative Element of REAL
sqr (normsequence (G,p1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
Sum (sqr (normsequence (G,p1))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (normsequence (G,p1)))) is V11() real ext-real Element of REAL
abs (1 - t) is V11() real ext-real Element of REAL
(abs (1 - t)) * ||.(f - F).|| is V11() real ext-real Element of REAL
abs t is V11() real ext-real Element of REAL
(abs t) * ||.(X - F).|| is V11() real ext-real Element of REAL
((abs (1 - t)) * ||.(f - F).||) + ((abs t) * ||.(X - F).||) is V11() real ext-real Element of REAL
(1 - t) * i0 is V11() real ext-real Element of REAL
t * i0 is V11() real ext-real Element of REAL
((1 - t) * i0) + (t * i0) is V11() real ext-real Element of REAL
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
K6( the carrier of (product G)) is set
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of F is non empty non trivial set
K7( the carrier of (product G), the carrier of F) is Relation-like set
K6(K7( the carrier of (product G), the carrier of F)) is set
f is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of F))
dom f is Element of K6( the carrier of (product G))
X is Element of K6( the carrier of (product G))
m is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
i is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
i - m is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
- m is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
i + (- m) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
the addF of (product G) is Relation-like K7( the carrier of (product G), the carrier of (product G)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)))
K7( the carrier of (product G), the carrier of (product G)) is Relation-like set
K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)) is Relation-like set
K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G))) is set
K462( the carrier of (product G), the addF of (product G),i,(- m)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.(i - m).|| is V11() real ext-real non negative Element of REAL
the normF of (product G) is non empty Relation-like the carrier of (product G) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (product G),REAL))
K7( the carrier of (product G),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (product G),REAL)) is non trivial V49() set
the normF of (product G) . (i - m) is V11() real ext-real Element of REAL
i0 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
i0 - m is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
i0 + (- m) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),i0,(- m)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.(i0 - m).|| is V11() real ext-real non negative Element of REAL
the normF of (product G) . (i0 - m) is V11() real ext-real Element of REAL
f /. i0 is Element of the carrier of F
f /. i is Element of the carrier of F
(f /. i0) - (f /. i) is Element of the carrier of F
- (f /. i) is Element of the carrier of F
(f /. i0) + (- (f /. i)) is Element of the carrier of F
the addF of F is Relation-like K7( the carrier of F, the carrier of F) -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of F, the carrier of F), the carrier of F))
K7( the carrier of F, the carrier of F) is Relation-like set
K7(K7( the carrier of F, the carrier of F), the carrier of F) is Relation-like set
K6(K7(K7( the carrier of F, the carrier of F), the carrier of F)) is set
K462( the carrier of F, the addF of F,(f /. i0),(- (f /. i))) is Element of the carrier of F
y0 is set
(G,y0) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . (G,y0) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . (G,y0)) is non empty non trivial set
R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,y0)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,y0)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,y0)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,y0)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,y0)),F))
K266( the carrier of (G . (G,y0)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,y0)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,y0)), the carrier of F
FuncZero ( the carrier of (G . (G,y0)),F) is Relation-like the carrier of (G . (G,y0)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,y0)), the carrier of F)
0. F is V68(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
the carrier of (G . (G,y0)) --> (0. F) is non empty Relation-like the carrier of (G . (G,y0)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,y0)), the carrier of F))
K7( the carrier of (G . (G,y0)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,y0)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,y0)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,y0)), the carrier of F)),(Funcs ( the carrier of (G . (G,y0)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,y0)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,y0)), the carrier of F)),(Funcs ( the carrier of (G . (G,y0)), the carrier of F))),(Funcs ( the carrier of (G . (G,y0)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,y0)), the carrier of F)),(Funcs ( the carrier of (G . (G,y0)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,y0)), the carrier of F)),(Funcs ( the carrier of (G . (G,y0)), the carrier of F))),(Funcs ( the carrier of (G . (G,y0)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,y0)), the carrier of F)),(Funcs ( the carrier of (G . (G,y0)), the carrier of F))),(Funcs ( the carrier of (G . (G,y0)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,y0)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,y0)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,y0)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,y0)), the carrier of F))),(Funcs ( the carrier of (G . (G,y0)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,y0)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,y0)), the carrier of F))),(Funcs ( the carrier of (G . (G,y0)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,y0)), the carrier of F))),(Funcs ( the carrier of (G . (G,y0)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,y0)), the carrier of F)),(FuncZero ( the carrier of (G . (G,y0)),F)),(FuncAdd ( the carrier of (G . (G,y0)),F)),(FuncExtMult ( the carrier of (G . (G,y0)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,y0)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,y0)),F)) is set
K215(K266( the carrier of (G . (G,y0)),F),(LinearOperators ((G . (G,y0)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,y0)),F)
K213(K266( the carrier of (G . (G,y0)),F),(LinearOperators ((G . (G,y0)),F))) is Relation-like K7((LinearOperators ((G . (G,y0)),F)),(LinearOperators ((G . (G,y0)),F))) -defined LinearOperators ((G . (G,y0)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,y0)),F)),(LinearOperators ((G . (G,y0)),F))),(LinearOperators ((G . (G,y0)),F))))
K7((LinearOperators ((G . (G,y0)),F)),(LinearOperators ((G . (G,y0)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,y0)),F)),(LinearOperators ((G . (G,y0)),F))),(LinearOperators ((G . (G,y0)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,y0)),F)),(LinearOperators ((G . (G,y0)),F))),(LinearOperators ((G . (G,y0)),F)))) is set
K214(K266( the carrier of (G . (G,y0)),F),(LinearOperators ((G . (G,y0)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,y0)),F))) -defined LinearOperators ((G . (G,y0)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,y0)),F))),(LinearOperators ((G . (G,y0)),F))))
K7(REAL,(LinearOperators ((G . (G,y0)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,y0)),F))),(LinearOperators ((G . (G,y0)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,y0)),F))),(LinearOperators ((G . (G,y0)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,y0)),F)),K215(K266( the carrier of (G . (G,y0)),F),(LinearOperators ((G . (G,y0)),F))),K213(K266( the carrier of (G . (G,y0)),F),(LinearOperators ((G . (G,y0)),F))),K214(K266( the carrier of (G . (G,y0)),F),(LinearOperators ((G . (G,y0)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,y0)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,y0)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))) is Element of BoundedLinearOperators ((G . (G,y0)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))) -defined BoundedLinearOperators ((G . (G,y0)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))),(BoundedLinearOperators ((G . (G,y0)),F))))
K7((BoundedLinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))),(BoundedLinearOperators ((G . (G,y0)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))),(BoundedLinearOperators ((G . (G,y0)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,y0)),F))) -defined BoundedLinearOperators ((G . (G,y0)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,y0)),F))),(BoundedLinearOperators ((G . (G,y0)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,y0)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,y0)),F))),(BoundedLinearOperators ((G . (G,y0)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,y0)),F))),(BoundedLinearOperators ((G . (G,y0)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,y0)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,y0)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,y0)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,y0)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,y0)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,y0)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,y0)),F)),(BoundedLinearOperators ((G . (G,y0)),F))),(BoundedLinearOperatorsNorm ((G . (G,y0)),F)) #) is strict NORMSTR
(G,F,y0,f,m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F))
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F)) is non empty set
(G,(G,y0),m) is non empty Relation-like the carrier of (G . (G,y0)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,y0)), the carrier of (product G)))
K7( the carrier of (G . (G,y0)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,y0)), the carrier of (product G))) is set
f * (G,(G,y0),m) is Relation-like the carrier of (G . (G,y0)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,y0)), the carrier of F))
(G,(G,y0)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,y0)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,y0))))
K7( the carrier of (product G), the carrier of (G . (G,y0))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,y0)))) is set
(G,(G,y0)) . m is Element of the carrier of (G . (G,y0))
diff ((f * (G,(G,y0),m)),((G,(G,y0)) . m)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F))
(G,(G,y0),i) is non empty Relation-like the carrier of (G . (G,y0)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,y0)), the carrier of (product G)))
(G,(G,y0)) . i is Element of the carrier of (G . (G,y0))
r is Element of the carrier of (G . (G,y0))
(G,(G,y0),i) . r is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
s is Element of the carrier of (G . (G,y0))
r - s is Element of the carrier of (G . (G,y0))
- s is Element of the carrier of (G . (G,y0))
r + (- s) is Element of the carrier of (G . (G,y0))
the addF of (G . (G,y0)) is Relation-like K7( the carrier of (G . (G,y0)), the carrier of (G . (G,y0))) -defined the carrier of (G . (G,y0)) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . (G,y0)), the carrier of (G . (G,y0))), the carrier of (G . (G,y0))))
K7( the carrier of (G . (G,y0)), the carrier of (G . (G,y0))) is Relation-like set
K7(K7( the carrier of (G . (G,y0)), the carrier of (G . (G,y0))), the carrier of (G . (G,y0))) is Relation-like set
K6(K7(K7( the carrier of (G . (G,y0)), the carrier of (G . (G,y0))), the carrier of (G . (G,y0)))) is set
K462( the carrier of (G . (G,y0)), the addF of (G . (G,y0)),r,(- s)) is Element of the carrier of (G . (G,y0))
(G,F,y0,f,m) . (r - s) is Element of the carrier of F
((f /. i0) - (f /. i)) - ((G,F,y0,f,m) . (r - s)) is Element of the carrier of F
- ((G,F,y0,f,m) . (r - s)) is Element of the carrier of F
((f /. i0) - (f /. i)) + (- ((G,F,y0,f,m) . (r - s))) is Element of the carrier of F
K462( the carrier of F, the addF of F,((f /. i0) - (f /. i)),(- ((G,F,y0,f,m) . (r - s)))) is Element of the carrier of F
||.(((f /. i0) - (f /. i)) - ((G,F,y0,f,m) . (r - s))).|| is V11() real ext-real non negative Element of REAL
the normF of F is non empty Relation-like the carrier of F -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of F,REAL))
K7( the carrier of F,REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of F,REAL)) is non trivial V49() set
the normF of F . (((f /. i0) - (f /. i)) - ((G,F,y0,f,m) . (r - s))) is V11() real ext-real Element of REAL
||.(r - s).|| is V11() real ext-real non negative Element of REAL
the normF of (G . (G,y0)) is non empty Relation-like the carrier of (G . (G,y0)) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (G . (G,y0)),REAL))
K7( the carrier of (G . (G,y0)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (G . (G,y0)),REAL)) is non trivial V49() set
the normF of (G . (G,y0)) . (r - 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
||.(r - s).|| * PP is V11() real ext-real Element of REAL
(G,(G,y0),i) . s is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
a is Element of the carrier of (G . (G,y0))
LSeg (s,r) is Element of K6( the carrier of (G . (G,y0)))
K6( the carrier of (G . (G,y0))) is set
(G,(G,y0),i) . a is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((G,(G,y0),i) . a) - m is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((G,(G,y0),i) . a) + (- m) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),((G,(G,y0),i) . a),(- m)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.(((G,(G,y0),i) . a) - m).|| is V11() real ext-real non negative Element of REAL
the normF of (product G) . (((G,(G,y0),i) . a) - m) is V11() real ext-real Element of REAL
a is Element of the carrier of (G . (G,y0))
(G,(G,y0),i) . a is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((G,(G,y0),i) . a) - m is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((G,(G,y0),i) . a) + (- m) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),((G,(G,y0),i) . a),(- m)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.(((G,(G,y0),i) . a) - m).|| is V11() real ext-real non negative Element of REAL
the normF of (product G) . (((G,(G,y0),i) . a) - m) is V11() real ext-real Element of REAL
((G . (G,y0)),s,r) is Element of K6( the carrier of (G . (G,y0)))
{ (s + (b₁ * (r - s))) where b₁ is V11() real ext-real Element of REAL : ( not b₁ <= 0 & not 1 <= b₁ ) } is set
a is Element of the carrier of (G . (G,y0))
(G,(G,y0),i) . a is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,F,y0,f,((G,(G,y0),i) . a)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F))
(G,(G,y0),((G,(G,y0),i) . a)) is non empty Relation-like the carrier of (G . (G,y0)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,y0)), the carrier of (product G)))
f * (G,(G,y0),((G,(G,y0),i) . a)) is Relation-like the carrier of (G . (G,y0)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,y0)), the carrier of F))
(G,(G,y0)) . ((G,(G,y0),i) . a) is Element of the carrier of (G . (G,y0))
diff ((f * (G,(G,y0),((G,(G,y0),i) . a))),((G,(G,y0)) . ((G,(G,y0),i) . a))) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F))
(G,F,y0,f,((G,(G,y0),i) . a)) - (G,F,y0,f,m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F))
- (G,F,y0,f,m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F))
(G,F,y0,f,((G,(G,y0),i) . a)) + (- (G,F,y0,f,m)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F))
the addF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F)) is Relation-like K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F))) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F)) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F))))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F))) is Relation-like set
K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F))) is Relation-like set
K6(K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F)))) is set
K462( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F)), the addF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F)),(G,F,y0,f,((G,(G,y0),i) . a)),(- (G,F,y0,f,m))) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F))
||.((G,F,y0,f,((G,(G,y0),i) . a)) - (G,F,y0,f,m)).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F)) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F)) -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 ((G . (G,y0)),F)),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F)),REAL)) is non trivial V49() set
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,y0)),F)) . ((G,F,y0,f,((G,(G,y0),i) . a)) - (G,F,y0,f,m)) is V11() real ext-real Element of REAL
((G,(G,y0),i) . a) - m is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((G,(G,y0),i) . a) + (- m) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),((G,(G,y0),i) . a),(- m)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.(((G,(G,y0),i) . a) - m).|| is V11() real ext-real non negative Element of REAL
the normF of (product G) . (((G,(G,y0),i) . a) - m) is V11() real ext-real Element of REAL
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
0. (product G) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like V68( product G) countable Element of the carrier of (product G)
the ZeroF of (product G) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(len G) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
F is Relation-like NAT -defined the carrier of (product G) -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of (product G)
len F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
dom F is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
f is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
X is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
X + f is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
the addF of (product G) is Relation-like K7( the carrier of (product G), the carrier of (product G)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)))
K7( the carrier of (product G), the carrier of (product G)) is Relation-like set
K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)) is Relation-like set
K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G))) is set
K462( the carrier of (product G), the addF of (product G),X,f) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
m is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
i is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
i0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
F /. i0 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
X + (F /. i0) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),X,(F /. i0)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((len G) + 1) -' i0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
(G,(((len G) + 1) -' i0)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
G . (G,(((len G) + 1) -' i0)) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . (G,(((len G) + 1) -' i0))) is non empty non trivial set
i0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
F /. (i0 + 1) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
X + (F /. (i0 + 1)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),X,(F /. (i0 + 1))) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,(G,(((len G) + 1) -' i0)),(X + (F /. (i0 + 1)))) is non empty Relation-like the carrier of (G . (G,(((len G) + 1) -' i0))) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,(((len G) + 1) -' i0))), the carrier of (product G)))
K7( the carrier of (G . (G,(((len G) + 1) -' i0))), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,(((len G) + 1) -' i0))), the carrier of (product G))) is set
(G,(G,(((len G) + 1) -' i0))) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,(((len G) + 1) -' i0))) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,(((len G) + 1) -' i0)))))
K7( the carrier of (product G), the carrier of (G . (G,(((len G) + 1) -' i0)))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,(((len G) + 1) -' i0))))) is set
(G,(G,(((len G) + 1) -' i0))) . (X + f) is Element of the carrier of (G . (G,(((len G) + 1) -' i0)))
(G,(G,(((len G) + 1) -' i0)),(X + (F /. (i0 + 1)))) . ((G,(G,(((len G) + 1) -' i0))) . (X + f)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
Seg (len F) is V49() len F -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= len F ) } is set
Seg (((len G) + 1) -' i0) is V49() ((len G) + 1) -' i0 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= ((len G) + 1) -' i0 ) } is set
m | (Seg (((len G) + 1) -' i0)) is Relation-like NAT -defined Seg (((len G) + 1) -' i0) -defined NAT -defined Function-like carr G -compatible V49() countable Element of sproduct (carr G)
sproduct (carr G) is non empty functional set
i +* (m | (Seg (((len G) + 1) -' i0))) is Relation-like NAT -defined Function-like carr G -compatible set
((len G) + 1) -' (i0 + 1) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
Seg (((len G) + 1) -' (i0 + 1)) is V49() ((len G) + 1) -' (i0 + 1) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= ((len G) + 1) -' (i0 + 1) ) } is set
m | (Seg (((len G) + 1) -' (i0 + 1))) is Relation-like NAT -defined Seg (((len G) + 1) -' (i0 + 1)) -defined NAT -defined Function-like carr G -compatible V49() countable Element of sproduct (carr G)
i +* (m | (Seg (((len G) + 1) -' (i0 + 1)))) is Relation-like NAT -defined Function-like carr G -compatible set
Seg (len G) is non empty V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= len G ) } is set
(len G) -' i0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
((len G) -' i0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
dom (carr G) is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
PP is Relation-like Function-like set
dom PP is set
dom (X + (F /. (i0 + 1))) is V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
y1 is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
y1 . (((len G) + 1) -' i0) is set
Replace ((G,(((len G) + 1) -' i0)),(y1 . (((len G) + 1) -' i0)),) is Relation-like Function-like set
(G,(((len G) + 1) -' i0)) .--> (y1 . (((len G) + 1) -' i0)) is trivial Relation-like dom G -defined {(G,(((len G) + 1) -' i0))} -defined Function-like one-to-one constant V49() countable set
{(G,(((len G) + 1) -' i0))} is non empty trivial V49() V53() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded set
(X + (F /. (i0 + 1))) +* ((G,(((len G) + 1) -' i0)) .--> (y1 . (((len G) + 1) -' i0))) is Relation-like Function-like V49() countable set
{(((len G) + 1) -' i0)} is non empty trivial V49() V53() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(REAL)
{(((len G) + 1) -' i0)} --> (y1 . (((len G) + 1) -' i0)) is non empty Relation-like {(((len G) + 1) -' i0)} -defined {(y1 . (((len G) + 1) -' i0))} -valued Function-like constant total quasi_total V49() countable Element of K6(K7({(((len G) + 1) -' i0)},{(y1 . (((len G) + 1) -' i0))}))
{(y1 . (((len G) + 1) -' i0))} is non empty trivial V49() 1 -element countable set
K7({(((len G) + 1) -' i0)},{(y1 . (((len G) + 1) -' i0))}) is Relation-like V49() countable set
K6(K7({(((len G) + 1) -' i0)},{(y1 . (((len G) + 1) -' i0))})) is V49() V53() countable set
(X + (F /. (i0 + 1))) +* ({(((len G) + 1) -' i0)} --> (y1 . (((len G) + 1) -' i0))) is non empty Relation-like Function-like V49() countable set
DD is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable set
len DD is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
a is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable set
len a is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
w2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
DD . w2 is set
a . w2 is set
Seg (len DD) is V49() len DD -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= len DD ) } is set
ys is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
(G,ys) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . ys) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . ys)))
G . ys is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . ys) is non empty non trivial set
K7( the carrier of (product G), the carrier of (G . ys)) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . ys))) is set
tm is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
(G,ys) . tm is set
(G,ys) . X is Element of the carrier of (G . ys)
(G,ys) . (F /. i0) is Element of the carrier of (G . ys)
((G,ys) . X) + ((G,ys) . (F /. i0)) is Element of the carrier of (G . ys)
the addF of (G . ys) is Relation-like K7( the carrier of (G . ys), the carrier of (G . ys)) -defined the carrier of (G . ys) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . ys), the carrier of (G . ys)), the carrier of (G . ys)))
K7( the carrier of (G . ys), the carrier of (G . ys)) is Relation-like set
K7(K7( the carrier of (G . ys), the carrier of (G . ys)), the carrier of (G . ys)) is Relation-like set
K6(K7(K7( the carrier of (G . ys), the carrier of (G . ys)), the carrier of (G . ys))) is set
K462( the carrier of (G . ys), the addF of (G . ys),((G,ys) . X),((G,ys) . (F /. i0))) is Element of the carrier of (G . ys)
dom m is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below Element of K6(NAT)
t is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
t . w2 is set
dom (m | (Seg (((len G) + 1) -' i0))) is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
(dom m) /\ (Seg (((len G) + 1) -' i0)) is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
i . w2 is set
(G,ys) . (0. (product G)) is Element of the carrier of (G . ys)
dom ({(((len G) + 1) -' i0)} --> (y1 . (((len G) + 1) -' i0))) is non empty V49() V53() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6({(((len G) + 1) -' i0)})
K6({(((len G) + 1) -' i0)}) is V49() V53() countable set
((X + (F /. (i0 + 1))) +* ({(((len G) + 1) -' i0)} --> (y1 . (((len G) + 1) -' i0)))) . ys is set
(X + (F /. (i0 + 1))) . ys is set
(G,ys) . (X + (F /. (i0 + 1))) is Element of the carrier of (G . ys)
(G,ys) . (F /. (i0 + 1)) is Element of the carrier of (G . ys)
Sw1 is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
Sw1 . w2 is set
(dom m) /\ (Seg (((len G) + 1) -' (i0 + 1))) is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
dom (m | (Seg (((len G) + 1) -' (i0 + 1)))) is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
(i +* (m | (Seg (((len G) + 1) -' (i0 + 1))))) . w2 is set
(m | (Seg (((len G) + 1) -' i0))) . w2 is set
m . w2 is set
(G,ys) . f is Element of the carrier of (G . ys)
(G,ys) . (X + f) is Element of the carrier of (G . ys)
{w2} is non empty trivial V49() V53() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(REAL)
y1 . w2 is set
{w2} --> (y1 . w2) is non empty Relation-like {w2} -defined {(y1 . w2)} -valued Function-like constant total quasi_total V49() countable Element of K6(K7({w2},{(y1 . w2)}))
{(y1 . w2)} is non empty trivial V49() 1 -element countable set
K7({w2},{(y1 . w2)}) is Relation-like V49() countable set
K6(K7({w2},{(y1 . w2)})) is V49() V53() countable set
dom ({w2} --> (y1 . w2)) is non empty V49() V53() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6({w2})
K6({w2}) is V49() V53() countable set
(X + (F /. (i0 + 1))) +* ({w2} --> (y1 . w2)) is non empty Relation-like Function-like V49() countable set
((X + (F /. (i0 + 1))) +* ({w2} --> (y1 . w2))) . ys is set
({w2} --> (y1 . w2)) . w2 is set
dom ({(((len G) + 1) -' i0)} --> (y1 . (((len G) + 1) -' i0))) is non empty V49() V53() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6({(((len G) + 1) -' i0)})
K6({(((len G) + 1) -' i0)}) is V49() V53() countable set
(X + (F /. (i0 + 1))) . w2 is set
(G,ys) . (X + (F /. (i0 + 1))) is Element of the carrier of (G . ys)
(((len G) + 1) -' i0) -' 1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
dom (m | (Seg (((len G) + 1) -' (i0 + 1)))) is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
(G,ys) . (F /. (i0 + 1)) is Element of the carrier of (G . ys)
Sw1 is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
Sw1 . w2 is set
(m | (Seg (((len G) + 1) -' (i0 + 1)))) . ys is set
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
0. (product G) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like V68( product G) countable Element of the carrier of (product G)
the ZeroF of (product G) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(len G) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
F is Relation-like NAT -defined the carrier of (product G) -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of (product G)
len F is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
dom F is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
f is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
X is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
X + f is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
the addF of (product G) is Relation-like K7( the carrier of (product G), the carrier of (product G)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)))
K7( the carrier of (product G), the carrier of (product G)) is Relation-like set
K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)) is Relation-like set
K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G))) is set
K462( the carrier of (product G), the addF of (product G),X,f) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
m is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
i is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
i0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
((len G) + 1) -' i0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
(G,(((len G) + 1) -' i0)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
G . (G,(((len G) + 1) -' i0)) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . (G,(((len G) + 1) -' i0))) is non empty non trivial set
(G,(G,(((len G) + 1) -' i0))) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,(((len G) + 1) -' i0))) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,(((len G) + 1) -' i0)))))
K7( the carrier of (product G), the carrier of (G . (G,(((len G) + 1) -' i0)))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,(((len G) + 1) -' i0))))) is set
(G,(G,(((len G) + 1) -' i0))) . (X + f) is Element of the carrier of (G . (G,(((len G) + 1) -' i0)))
i0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
F /. (i0 + 1) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
X + (F /. (i0 + 1)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),X,(F /. (i0 + 1))) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,(G,(((len G) + 1) -' i0))) . (X + (F /. (i0 + 1))) is Element of the carrier of (G . (G,(((len G) + 1) -' i0)))
((G,(G,(((len G) + 1) -' i0))) . (X + f)) - ((G,(G,(((len G) + 1) -' i0))) . (X + (F /. (i0 + 1)))) is Element of the carrier of (G . (G,(((len G) + 1) -' i0)))
- ((G,(G,(((len G) + 1) -' i0))) . (X + (F /. (i0 + 1)))) is Element of the carrier of (G . (G,(((len G) + 1) -' i0)))
((G,(G,(((len G) + 1) -' i0))) . (X + f)) + (- ((G,(G,(((len G) + 1) -' i0))) . (X + (F /. (i0 + 1))))) is Element of the carrier of (G . (G,(((len G) + 1) -' i0)))
the addF of (G . (G,(((len G) + 1) -' i0))) is Relation-like K7( the carrier of (G . (G,(((len G) + 1) -' i0))), the carrier of (G . (G,(((len G) + 1) -' i0)))) -defined the carrier of (G . (G,(((len G) + 1) -' i0))) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . (G,(((len G) + 1) -' i0))), the carrier of (G . (G,(((len G) + 1) -' i0)))), the carrier of (G . (G,(((len G) + 1) -' i0)))))
K7( the carrier of (G . (G,(((len G) + 1) -' i0))), the carrier of (G . (G,(((len G) + 1) -' i0)))) is Relation-like set
K7(K7( the carrier of (G . (G,(((len G) + 1) -' i0))), the carrier of (G . (G,(((len G) + 1) -' i0)))), the carrier of (G . (G,(((len G) + 1) -' i0)))) is Relation-like set
K6(K7(K7( the carrier of (G . (G,(((len G) + 1) -' i0))), the carrier of (G . (G,(((len G) + 1) -' i0)))), the carrier of (G . (G,(((len G) + 1) -' i0))))) is set
K462( the carrier of (G . (G,(((len G) + 1) -' i0))), the addF of (G . (G,(((len G) + 1) -' i0))),((G,(G,(((len G) + 1) -' i0))) . (X + f)),(- ((G,(G,(((len G) + 1) -' i0))) . (X + (F /. (i0 + 1)))))) is Element of the carrier of (G . (G,(((len G) + 1) -' i0)))
(G,(G,(((len G) + 1) -' i0))) . f is Element of the carrier of (G . (G,(((len G) + 1) -' i0)))
F /. i0 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
X + (F /. i0) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),X,(F /. i0)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,(G,(((len G) + 1) -' i0)),(X + (F /. (i0 + 1)))) is non empty Relation-like the carrier of (G . (G,(((len G) + 1) -' i0))) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,(((len G) + 1) -' i0))), the carrier of (product G)))
K7( the carrier of (G . (G,(((len G) + 1) -' i0))), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,(((len G) + 1) -' i0))), the carrier of (product G))) is set
(G,(G,(((len G) + 1) -' i0)),(X + (F /. (i0 + 1)))) . ((G,(G,(((len G) + 1) -' i0))) . (X + f)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,(G,(((len G) + 1) -' i0))) . (X + (F /. i0)) is Element of the carrier of (G . (G,(((len G) + 1) -' i0)))
(X + (F /. i0)) - (X + (F /. (i0 + 1))) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
- (X + (F /. (i0 + 1))) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(X + (F /. i0)) + (- (X + (F /. (i0 + 1)))) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),(X + (F /. i0)),(- (X + (F /. (i0 + 1))))) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,(G,(((len G) + 1) -' i0))) . ((X + (F /. i0)) - (X + (F /. (i0 + 1)))) is Element of the carrier of (G . (G,(((len G) + 1) -' i0)))
(F /. i0) + X is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),(F /. i0),X) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((F /. i0) + X) - X is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
- X is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((F /. i0) + X) + (- X) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),((F /. i0) + X),(- X)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(((F /. i0) + X) - X) - (F /. (i0 + 1)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
- (F /. (i0 + 1)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(((F /. i0) + X) - X) + (- (F /. (i0 + 1))) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),(((F /. i0) + X) - X),(- (F /. (i0 + 1)))) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
X - X is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
X + (- X) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),X,(- X)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(F /. i0) + (X - X) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),(F /. i0),(X - X)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((F /. i0) + (X - X)) - (F /. (i0 + 1)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((F /. i0) + (X - X)) + (- (F /. (i0 + 1))) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),((F /. i0) + (X - X)),(- (F /. (i0 + 1)))) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(F /. i0) + (0. (product G)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),(F /. i0),(0. (product G))) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((F /. i0) + (0. (product G))) - (F /. (i0 + 1)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
((F /. i0) + (0. (product G))) + (- (F /. (i0 + 1))) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),((F /. i0) + (0. (product G))),(- (F /. (i0 + 1)))) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(F /. i0) - (F /. (i0 + 1)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(F /. i0) + (- (F /. (i0 + 1))) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),(F /. i0),(- (F /. (i0 + 1)))) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,(G,(((len G) + 1) -' i0))) . (F /. i0) is Element of the carrier of (G . (G,(((len G) + 1) -' i0)))
(G,(G,(((len G) + 1) -' i0))) . (F /. (i0 + 1)) is Element of the carrier of (G . (G,(((len G) + 1) -' i0)))
((G,(G,(((len G) + 1) -' i0))) . (F /. i0)) - ((G,(G,(((len G) + 1) -' i0))) . (F /. (i0 + 1))) is Element of the carrier of (G . (G,(((len G) + 1) -' i0)))
- ((G,(G,(((len G) + 1) -' i0))) . (F /. (i0 + 1))) is Element of the carrier of (G . (G,(((len G) + 1) -' i0)))
((G,(G,(((len G) + 1) -' i0))) . (F /. i0)) + (- ((G,(G,(((len G) + 1) -' i0))) . (F /. (i0 + 1)))) is Element of the carrier of (G . (G,(((len G) + 1) -' i0)))
K462( the carrier of (G . (G,(((len G) + 1) -' i0))), the addF of (G . (G,(((len G) + 1) -' i0))),((G,(G,(((len G) + 1) -' i0))) . (F /. i0)),(- ((G,(G,(((len G) + 1) -' i0))) . (F /. (i0 + 1))))) is Element of the carrier of (G . (G,(((len G) + 1) -' i0)))
dom m is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below Element of K6(NAT)
Seg (len G) is non empty V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= len G ) } is set
(len G) -' i0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
((len G) -' i0) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (((len G) + 1) -' i0) is V49() ((len G) + 1) -' i0 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= ((len G) + 1) -' i0 ) } is set
m | (Seg (((len G) + 1) -' i0)) is Relation-like NAT -defined Seg (((len G) + 1) -' i0) -defined NAT -defined Function-like carr G -compatible V49() countable Element of sproduct (carr G)
sproduct (carr G) is non empty functional set
dom (m | (Seg (((len G) + 1) -' i0))) is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
((len G) + 1) -' (i0 + 1) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
(((len G) + 1) -' (i0 + 1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (((len G) + 1) -' (i0 + 1)) is V49() ((len G) + 1) -' (i0 + 1) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= ((len G) + 1) -' (i0 + 1) ) } is set
m | (Seg (((len G) + 1) -' (i0 + 1))) is Relation-like NAT -defined Seg (((len G) + 1) -' (i0 + 1)) -defined NAT -defined Function-like carr G -compatible V49() countable Element of sproduct (carr G)
dom (m | (Seg (((len G) + 1) -' (i0 + 1)))) is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
Seg (len F) is V49() len F -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= len F ) } is set
i +* (m | (Seg (((len G) + 1) -' i0))) is Relation-like NAT -defined Function-like carr G -compatible set
y0 is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
y0 . (((len G) + 1) -' i0) is set
(m | (Seg (((len G) + 1) -' i0))) . (((len G) + 1) -' i0) is set
m . (((len G) + 1) -' i0) is set
i +* (m | (Seg (((len G) + 1) -' (i0 + 1)))) is Relation-like NAT -defined Function-like carr G -compatible set
r is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
r . (((len G) + 1) -' i0) is set
i . (((len G) + 1) -' i0) is set
(G,(G,(((len G) + 1) -' i0))) . (0. (product G)) is Element of the carrier of (G . (G,(((len G) + 1) -' i0)))
f - (0. (product G)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
- (0. (product G)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f + (- (0. (product G))) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),f,(- (0. (product G)))) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,(G,(((len G) + 1) -' i0))) . (f - (0. (product G))) is Element of the carrier of (G . (G,(((len G) + 1) -' i0)))
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
K6( the carrier of (product G)) is set
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of F is non empty non trivial set
K7( the carrier of (product G), the carrier of F) is Relation-like set
K6(K7( the carrier of (product G), the carrier of F)) is set
f is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of F))
X is Element of K6( the carrier of (product G))
m is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
diff (f,m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
R_NormSpace_of_BoundedLinearOperators ((product G),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((product G),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((product G),F)))
R_VectorSpace_of_LinearOperators ((product G),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((product G),F) is non empty functional Element of K6( the carrier of K266( the carrier of (product G),F))
K266( the carrier of (product G),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (product G), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (product G), the carrier of F
FuncZero ( the carrier of (product G),F) is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (product G), the carrier of F)
0. F is V68(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
the carrier of (product G) --> (0. F) is non empty Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (product G), the carrier of F))
FuncAdd ( the carrier of (product G),F) is Relation-like K7((Funcs ( the carrier of (product G), the carrier of F)),(Funcs ( the carrier of (product G), the carrier of F))) -defined Funcs ( the carrier of (product G), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (product G), the carrier of F)),(Funcs ( the carrier of (product G), the carrier of F))),(Funcs ( the carrier of (product G), the carrier of F))))
K7((Funcs ( the carrier of (product G), the carrier of F)),(Funcs ( the carrier of (product G), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (product G), the carrier of F)),(Funcs ( the carrier of (product G), the carrier of F))),(Funcs ( the carrier of (product G), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (product G), the carrier of F)),(Funcs ( the carrier of (product G), the carrier of F))),(Funcs ( the carrier of (product G), the carrier of F)))) is set
FuncExtMult ( the carrier of (product G),F) is Relation-like K7(REAL,(Funcs ( the carrier of (product G), the carrier of F))) -defined Funcs ( the carrier of (product G), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (product G), the carrier of F))),(Funcs ( the carrier of (product G), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (product G), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (product G), the carrier of F))),(Funcs ( the carrier of (product G), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (product G), the carrier of F))),(Funcs ( the carrier of (product G), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (product G), the carrier of F)),(FuncZero ( the carrier of (product G),F)),(FuncAdd ( the carrier of (product G),F)),(FuncExtMult ( the carrier of (product G),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (product G),F) is non empty set
K6( the carrier of K266( the carrier of (product G),F)) is set
K215(K266( the carrier of (product G),F),(LinearOperators ((product G),F))) is Relation-like Function-like Element of LinearOperators ((product G),F)
K213(K266( the carrier of (product G),F),(LinearOperators ((product G),F))) is Relation-like K7((LinearOperators ((product G),F)),(LinearOperators ((product G),F))) -defined LinearOperators ((product G),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((product G),F)),(LinearOperators ((product G),F))),(LinearOperators ((product G),F))))
K7((LinearOperators ((product G),F)),(LinearOperators ((product G),F))) is Relation-like set
K7(K7((LinearOperators ((product G),F)),(LinearOperators ((product G),F))),(LinearOperators ((product G),F))) is Relation-like set
K6(K7(K7((LinearOperators ((product G),F)),(LinearOperators ((product G),F))),(LinearOperators ((product G),F)))) is set
K214(K266( the carrier of (product G),F),(LinearOperators ((product G),F))) is Relation-like K7(REAL,(LinearOperators ((product G),F))) -defined LinearOperators ((product G),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((product G),F))),(LinearOperators ((product G),F))))
K7(REAL,(LinearOperators ((product G),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((product G),F))),(LinearOperators ((product G),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((product G),F))),(LinearOperators ((product G),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((product G),F)),K215(K266( the carrier of (product G),F),(LinearOperators ((product G),F))),K213(K266( the carrier of (product G),F),(LinearOperators ((product G),F))),K214(K266( the carrier of (product G),F),(LinearOperators ((product G),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((product G),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((product G),F))) is set
K215((R_VectorSpace_of_LinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))) is Element of BoundedLinearOperators ((product G),F)
K213((R_VectorSpace_of_LinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))) is Relation-like K7((BoundedLinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))) -defined BoundedLinearOperators ((product G),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))),(BoundedLinearOperators ((product G),F))))
K7((BoundedLinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))),(BoundedLinearOperators ((product G),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))),(BoundedLinearOperators ((product G),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((product G),F))) -defined BoundedLinearOperators ((product G),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((product G),F))),(BoundedLinearOperators ((product G),F))))
K7(REAL,(BoundedLinearOperators ((product G),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((product G),F))),(BoundedLinearOperators ((product G),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((product G),F))),(BoundedLinearOperators ((product G),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((product G),F) is non empty Relation-like BoundedLinearOperators ((product G),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((product G),F)),REAL))
K7((BoundedLinearOperators ((product G),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((product G),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((product G),F)),K215((R_VectorSpace_of_LinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))),K213((R_VectorSpace_of_LinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))),K214((R_VectorSpace_of_LinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))),(BoundedLinearOperatorsNorm ((product G),F)) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)) is non empty set
Seg (len G) is non empty V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= len G ) } is set
carr G is non empty Relation-like non-empty NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
product (carr G) is non empty functional with_common_domain product-like set
0. (product G) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like V68( product G) countable Element of the carrier of (product G)
the ZeroF of (product G) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
y1 is non empty Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like total quasi_total V153( product G,F) V154( product G,F) Lipschitzian Element of K6(K7( the carrier of (product G), the carrier of F))
PP is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
y1 . PP is Element of the carrier of F
DD is Relation-like NAT -defined the carrier of F -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of F
dom DD is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
Sum DD is Element of the carrier of F
a is set
DD . a is set
(G,a) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . (G,a) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
(G,F,a,f,m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,a)),F))
R_NormSpace_of_BoundedLinearOperators ((G . (G,a)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,a)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,a)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,a)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,a)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,a)),F))
the carrier of (G . (G,a)) is non empty non trivial set
K266( the carrier of (G . (G,a)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,a)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,a)), the carrier of F
FuncZero ( the carrier of (G . (G,a)),F) is Relation-like the carrier of (G . (G,a)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,a)), the carrier of F)
the carrier of (G . (G,a)) --> (0. F) is non empty Relation-like the carrier of (G . (G,a)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,a)), the carrier of F))
K7( the carrier of (G . (G,a)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,a)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,a)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,a)), the carrier of F)),(Funcs ( the carrier of (G . (G,a)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,a)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,a)), the carrier of F)),(Funcs ( the carrier of (G . (G,a)), the carrier of F))),(Funcs ( the carrier of (G . (G,a)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,a)), the carrier of F)),(Funcs ( the carrier of (G . (G,a)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,a)), the carrier of F)),(Funcs ( the carrier of (G . (G,a)), the carrier of F))),(Funcs ( the carrier of (G . (G,a)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,a)), the carrier of F)),(Funcs ( the carrier of (G . (G,a)), the carrier of F))),(Funcs ( the carrier of (G . (G,a)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,a)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,a)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,a)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,a)), the carrier of F))),(Funcs ( the carrier of (G . (G,a)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,a)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,a)), the carrier of F))),(Funcs ( the carrier of (G . (G,a)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,a)), the carrier of F))),(Funcs ( the carrier of (G . (G,a)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,a)), the carrier of F)),(FuncZero ( the carrier of (G . (G,a)),F)),(FuncAdd ( the carrier of (G . (G,a)),F)),(FuncExtMult ( the carrier of (G . (G,a)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,a)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,a)),F)) is set
K215(K266( the carrier of (G . (G,a)),F),(LinearOperators ((G . (G,a)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,a)),F)
K213(K266( the carrier of (G . (G,a)),F),(LinearOperators ((G . (G,a)),F))) is Relation-like K7((LinearOperators ((G . (G,a)),F)),(LinearOperators ((G . (G,a)),F))) -defined LinearOperators ((G . (G,a)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,a)),F)),(LinearOperators ((G . (G,a)),F))),(LinearOperators ((G . (G,a)),F))))
K7((LinearOperators ((G . (G,a)),F)),(LinearOperators ((G . (G,a)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,a)),F)),(LinearOperators ((G . (G,a)),F))),(LinearOperators ((G . (G,a)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,a)),F)),(LinearOperators ((G . (G,a)),F))),(LinearOperators ((G . (G,a)),F)))) is set
K214(K266( the carrier of (G . (G,a)),F),(LinearOperators ((G . (G,a)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,a)),F))) -defined LinearOperators ((G . (G,a)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,a)),F))),(LinearOperators ((G . (G,a)),F))))
K7(REAL,(LinearOperators ((G . (G,a)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,a)),F))),(LinearOperators ((G . (G,a)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,a)),F))),(LinearOperators ((G . (G,a)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,a)),F)),K215(K266( the carrier of (G . (G,a)),F),(LinearOperators ((G . (G,a)),F))),K213(K266( the carrier of (G . (G,a)),F),(LinearOperators ((G . (G,a)),F))),K214(K266( the carrier of (G . (G,a)),F),(LinearOperators ((G . (G,a)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,a)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,a)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))) is Element of BoundedLinearOperators ((G . (G,a)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))) -defined BoundedLinearOperators ((G . (G,a)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))),(BoundedLinearOperators ((G . (G,a)),F))))
K7((BoundedLinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))),(BoundedLinearOperators ((G . (G,a)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))),(BoundedLinearOperators ((G . (G,a)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,a)),F))) -defined BoundedLinearOperators ((G . (G,a)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,a)),F))),(BoundedLinearOperators ((G . (G,a)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,a)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,a)),F))),(BoundedLinearOperators ((G . (G,a)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,a)),F))),(BoundedLinearOperators ((G . (G,a)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,a)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,a)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,a)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,a)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,a)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,a)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,a)),F)),(BoundedLinearOperators ((G . (G,a)),F))),(BoundedLinearOperatorsNorm ((G . (G,a)),F)) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,a)),F)) is non empty set
(G,(G,a),m) is non empty Relation-like the carrier of (G . (G,a)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,a)), the carrier of (product G)))
K7( the carrier of (G . (G,a)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,a)), the carrier of (product G))) is set
f * (G,(G,a),m) is Relation-like the carrier of (G . (G,a)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,a)), the carrier of F))
(G,(G,a)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,a)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,a))))
K7( the carrier of (product G), the carrier of (G . (G,a))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,a)))) is set
(G,(G,a)) . m is Element of the carrier of (G . (G,a))
diff ((f * (G,(G,a),m)),((G,(G,a)) . m)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,a)),F))
(G,(G,a)) . PP is Element of the carrier of (G . (G,a))
(G,F,a,f,m) . ((G,(G,a)) . PP) is Element of the carrier of F
PP is V11() real ext-real Element of REAL
{ b₁ where b₁ is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G) : not PP <= ||.(b₁ - m).|| } is set
dom f is Element of K6( the carrier of (product G))
f /. m is Element of the carrier of F
a is non empty Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of F))
t is V11() real ext-real Element of REAL
t / 2 is V11() real ext-real Element of REAL
K39(2) is non empty V11() real ext-real positive non negative set
K37(t,K39(2)) is V11() real ext-real set
(t / 2) / (len G) is V11() real ext-real Element of REAL
K39((len G)) is non empty V11() real ext-real positive non negative set
K37((t / 2),K39((len G))) is V11() real ext-real set
w2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
ys is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
(G,ys) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . (G,ys) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,ys)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,ys)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,ys)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,ys)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,ys)),F))
the carrier of (G . (G,ys)) is non empty non trivial set
K266( the carrier of (G . (G,ys)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,ys)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,ys)), the carrier of F
FuncZero ( the carrier of (G . (G,ys)),F) is Relation-like the carrier of (G . (G,ys)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,ys)), the carrier of F)
the carrier of (G . (G,ys)) --> (0. F) is non empty Relation-like the carrier of (G . (G,ys)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,ys)), the carrier of F))
K7( the carrier of (G . (G,ys)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,ys)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,ys)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,ys)), the carrier of F)),(Funcs ( the carrier of (G . (G,ys)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,ys)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,ys)), the carrier of F)),(Funcs ( the carrier of (G . (G,ys)), the carrier of F))),(Funcs ( the carrier of (G . (G,ys)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,ys)), the carrier of F)),(Funcs ( the carrier of (G . (G,ys)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,ys)), the carrier of F)),(Funcs ( the carrier of (G . (G,ys)), the carrier of F))),(Funcs ( the carrier of (G . (G,ys)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,ys)), the carrier of F)),(Funcs ( the carrier of (G . (G,ys)), the carrier of F))),(Funcs ( the carrier of (G . (G,ys)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,ys)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,ys)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,ys)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,ys)), the carrier of F))),(Funcs ( the carrier of (G . (G,ys)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,ys)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,ys)), the carrier of F))),(Funcs ( the carrier of (G . (G,ys)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,ys)), the carrier of F))),(Funcs ( the carrier of (G . (G,ys)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,ys)), the carrier of F)),(FuncZero ( the carrier of (G . (G,ys)),F)),(FuncAdd ( the carrier of (G . (G,ys)),F)),(FuncExtMult ( the carrier of (G . (G,ys)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,ys)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,ys)),F)) is set
K215(K266( the carrier of (G . (G,ys)),F),(LinearOperators ((G . (G,ys)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,ys)),F)
K213(K266( the carrier of (G . (G,ys)),F),(LinearOperators ((G . (G,ys)),F))) is Relation-like K7((LinearOperators ((G . (G,ys)),F)),(LinearOperators ((G . (G,ys)),F))) -defined LinearOperators ((G . (G,ys)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,ys)),F)),(LinearOperators ((G . (G,ys)),F))),(LinearOperators ((G . (G,ys)),F))))
K7((LinearOperators ((G . (G,ys)),F)),(LinearOperators ((G . (G,ys)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,ys)),F)),(LinearOperators ((G . (G,ys)),F))),(LinearOperators ((G . (G,ys)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,ys)),F)),(LinearOperators ((G . (G,ys)),F))),(LinearOperators ((G . (G,ys)),F)))) is set
K214(K266( the carrier of (G . (G,ys)),F),(LinearOperators ((G . (G,ys)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,ys)),F))) -defined LinearOperators ((G . (G,ys)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,ys)),F))),(LinearOperators ((G . (G,ys)),F))))
K7(REAL,(LinearOperators ((G . (G,ys)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,ys)),F))),(LinearOperators ((G . (G,ys)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,ys)),F))),(LinearOperators ((G . (G,ys)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,ys)),F)),K215(K266( the carrier of (G . (G,ys)),F),(LinearOperators ((G . (G,ys)),F))),K213(K266( the carrier of (G . (G,ys)),F),(LinearOperators ((G . (G,ys)),F))),K214(K266( the carrier of (G . (G,ys)),F),(LinearOperators ((G . (G,ys)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,ys)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,ys)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,ys)),F)),(BoundedLinearOperators ((G . (G,ys)),F))) is Element of BoundedLinearOperators ((G . (G,ys)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,ys)),F)),(BoundedLinearOperators ((G . (G,ys)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,ys)),F)),(BoundedLinearOperators ((G . (G,ys)),F))) -defined BoundedLinearOperators ((G . (G,ys)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,ys)),F)),(BoundedLinearOperators ((G . (G,ys)),F))),(BoundedLinearOperators ((G . (G,ys)),F))))
K7((BoundedLinearOperators ((G . (G,ys)),F)),(BoundedLinearOperators ((G . (G,ys)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,ys)),F)),(BoundedLinearOperators ((G . (G,ys)),F))),(BoundedLinearOperators ((G . (G,ys)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,ys)),F)),(BoundedLinearOperators ((G . (G,ys)),F))),(BoundedLinearOperators ((G . (G,ys)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,ys)),F)),(BoundedLinearOperators ((G . (G,ys)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,ys)),F))) -defined BoundedLinearOperators ((G . (G,ys)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,ys)),F))),(BoundedLinearOperators ((G . (G,ys)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,ys)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,ys)),F))),(BoundedLinearOperators ((G . (G,ys)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,ys)),F))),(BoundedLinearOperators ((G . (G,ys)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,ys)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,ys)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,ys)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,ys)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,ys)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,ys)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,ys)),F)),(BoundedLinearOperators ((G . (G,ys)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,ys)),F)),(BoundedLinearOperators ((G . (G,ys)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,ys)),F)),(BoundedLinearOperators ((G . (G,ys)),F))),(BoundedLinearOperatorsNorm ((G . (G,ys)),F)) #) is strict NORMSTR
(G,F,ys,f,X) is Relation-like the carrier of (product G) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F)) -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F))))
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F)) is non empty set
K7( the carrier of (product G), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F)))) is set
(G,F,ys,f,X) /. m is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F))
rm is V11() real ext-real Element of REAL
(G,F,ys,f,m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F))
(G,(G,ys),m) is non empty Relation-like the carrier of (G . (G,ys)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,ys)), the carrier of (product G)))
K7( the carrier of (G . (G,ys)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,ys)), the carrier of (product G))) is set
f * (G,(G,ys),m) is Relation-like the carrier of (G . (G,ys)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,ys)), the carrier of F))
(G,(G,ys)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,ys)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,ys))))
K7( the carrier of (product G), the carrier of (G . (G,ys))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,ys)))) is set
(G,(G,ys)) . m is Element of the carrier of (G . (G,ys))
diff ((f * (G,(G,ys),m)),((G,(G,ys)) . m)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F))
rs is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
rs - m is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
- m is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
rs + (- m) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
the addF of (product G) is Relation-like K7( the carrier of (product G), the carrier of (product G)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)))
K7( the carrier of (product G), the carrier of (product G)) is Relation-like set
K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)) is Relation-like set
K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G))) is set
K462( the carrier of (product G), the addF of (product G),rs,(- m)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.(rs - m).|| is V11() real ext-real non negative Element of REAL
the normF of (product G) is non empty Relation-like the carrier of (product G) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (product G),REAL))
K7( the carrier of (product G),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (product G),REAL)) is non trivial V49() set
the normF of (product G) . (rs - m) is V11() real ext-real Element of REAL
(G,F,ys,f,rs) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F))
(G,(G,ys),rs) is non empty Relation-like the carrier of (G . (G,ys)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,ys)), the carrier of (product G)))
f * (G,(G,ys),rs) is Relation-like the carrier of (G . (G,ys)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,ys)), the carrier of F))
(G,(G,ys)) . rs is Element of the carrier of (G . (G,ys))
diff ((f * (G,(G,ys),rs)),((G,(G,ys)) . rs)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F))
(G,F,ys,f,rs) - (G,F,ys,f,m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F))
- (G,F,ys,f,m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F))
(G,F,ys,f,rs) + (- (G,F,ys,f,m)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F))
the addF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F)) is Relation-like K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F))) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F)) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F))))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F))) is Relation-like set
K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F))) is Relation-like set
K6(K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F)))) is set
K462( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F)), the addF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F)),(G,F,ys,f,rs),(- (G,F,ys,f,m))) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F))
||.((G,F,ys,f,rs) - (G,F,ys,f,m)).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F)) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F)) -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 ((G . (G,ys)),F)),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F)),REAL)) is non trivial V49() set
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F)) . ((G,F,ys,f,rs) - (G,F,ys,f,m)) is V11() real ext-real Element of REAL
(G,F,ys,f,X) /. rs is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F))
((G,F,ys,f,X) /. rs) - ((G,F,ys,f,X) /. m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F))
- ((G,F,ys,f,X) /. m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F))
((G,F,ys,f,X) /. rs) + (- ((G,F,ys,f,X) /. m)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F))
K462( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F)), the addF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F)),((G,F,ys,f,X) /. rs),(- ((G,F,ys,f,X) /. m))) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F))
||.(((G,F,ys,f,X) /. rs) - ((G,F,ys,f,X) /. m)).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,ys)),F)) . (((G,F,ys,f,X) /. rs) - ((G,F,ys,f,X) /. m)) is V11() real ext-real Element of REAL
w2 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
dom w2 is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
rng w2 is V49() complex-membered ext-real-membered real-membered countable bounded_below bounded_above real-bounded Element of K6(REAL)
ys is non empty ext-real-membered set
rm is non empty ext-real-membered left_end right_end set
inf rm is ext-real set
rs is V11() real ext-real Element of REAL
min (PP,rs) is V11() real ext-real Element of REAL
p1 is set
w2 . p1 is V11() real ext-real Element of REAL
p0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
w2 . p0 is V11() real ext-real Element of REAL
P1 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
P1 - m is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
- m is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
P1 + (- m) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
the addF of (product G) is Relation-like K7( the carrier of (product G), the carrier of (product G)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)))
K7( the carrier of (product G), the carrier of (product G)) is Relation-like set
K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)) is Relation-like set
K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G))) is set
K462( the carrier of (product G), the addF of (product G),P1,(- m)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.(P1 - m).|| is V11() real ext-real non negative Element of REAL
the normF of (product G) is non empty Relation-like the carrier of (product G) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (product G),REAL))
K7( the carrier of (product G),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (product G),REAL)) is non trivial V49() set
the normF of (product G) . (P1 - m) is V11() real ext-real Element of REAL
P1 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
P1 - m is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
P1 + (- m) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),P1,(- m)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.(P1 - m).|| is V11() real ext-real non negative Element of REAL
the normF of (product G) . (P1 - m) is V11() real ext-real Element of REAL
P0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
PP is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
w2 . PP is V11() real ext-real Element of REAL
pt is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
(G,pt) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . (G,pt) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
R_NormSpace_of_BoundedLinearOperators ((G . (G,pt)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,pt)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,pt)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,pt)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,pt)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,pt)),F))
the carrier of (G . (G,pt)) is non empty non trivial set
K266( the carrier of (G . (G,pt)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,pt)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,pt)), the carrier of F
FuncZero ( the carrier of (G . (G,pt)),F) is Relation-like the carrier of (G . (G,pt)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,pt)), the carrier of F)
the carrier of (G . (G,pt)) --> (0. F) is non empty Relation-like the carrier of (G . (G,pt)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,pt)), the carrier of F))
K7( the carrier of (G . (G,pt)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,pt)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,pt)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,pt)), the carrier of F)),(Funcs ( the carrier of (G . (G,pt)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,pt)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,pt)), the carrier of F)),(Funcs ( the carrier of (G . (G,pt)), the carrier of F))),(Funcs ( the carrier of (G . (G,pt)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,pt)), the carrier of F)),(Funcs ( the carrier of (G . (G,pt)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,pt)), the carrier of F)),(Funcs ( the carrier of (G . (G,pt)), the carrier of F))),(Funcs ( the carrier of (G . (G,pt)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,pt)), the carrier of F)),(Funcs ( the carrier of (G . (G,pt)), the carrier of F))),(Funcs ( the carrier of (G . (G,pt)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,pt)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,pt)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,pt)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,pt)), the carrier of F))),(Funcs ( the carrier of (G . (G,pt)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,pt)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,pt)), the carrier of F))),(Funcs ( the carrier of (G . (G,pt)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,pt)), the carrier of F))),(Funcs ( the carrier of (G . (G,pt)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,pt)), the carrier of F)),(FuncZero ( the carrier of (G . (G,pt)),F)),(FuncAdd ( the carrier of (G . (G,pt)),F)),(FuncExtMult ( the carrier of (G . (G,pt)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,pt)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,pt)),F)) is set
K215(K266( the carrier of (G . (G,pt)),F),(LinearOperators ((G . (G,pt)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,pt)),F)
K213(K266( the carrier of (G . (G,pt)),F),(LinearOperators ((G . (G,pt)),F))) is Relation-like K7((LinearOperators ((G . (G,pt)),F)),(LinearOperators ((G . (G,pt)),F))) -defined LinearOperators ((G . (G,pt)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,pt)),F)),(LinearOperators ((G . (G,pt)),F))),(LinearOperators ((G . (G,pt)),F))))
K7((LinearOperators ((G . (G,pt)),F)),(LinearOperators ((G . (G,pt)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,pt)),F)),(LinearOperators ((G . (G,pt)),F))),(LinearOperators ((G . (G,pt)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,pt)),F)),(LinearOperators ((G . (G,pt)),F))),(LinearOperators ((G . (G,pt)),F)))) is set
K214(K266( the carrier of (G . (G,pt)),F),(LinearOperators ((G . (G,pt)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,pt)),F))) -defined LinearOperators ((G . (G,pt)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,pt)),F))),(LinearOperators ((G . (G,pt)),F))))
K7(REAL,(LinearOperators ((G . (G,pt)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,pt)),F))),(LinearOperators ((G . (G,pt)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,pt)),F))),(LinearOperators ((G . (G,pt)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,pt)),F)),K215(K266( the carrier of (G . (G,pt)),F),(LinearOperators ((G . (G,pt)),F))),K213(K266( the carrier of (G . (G,pt)),F),(LinearOperators ((G . (G,pt)),F))),K214(K266( the carrier of (G . (G,pt)),F),(LinearOperators ((G . (G,pt)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,pt)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,pt)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,pt)),F)),(BoundedLinearOperators ((G . (G,pt)),F))) is Element of BoundedLinearOperators ((G . (G,pt)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,pt)),F)),(BoundedLinearOperators ((G . (G,pt)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,pt)),F)),(BoundedLinearOperators ((G . (G,pt)),F))) -defined BoundedLinearOperators ((G . (G,pt)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,pt)),F)),(BoundedLinearOperators ((G . (G,pt)),F))),(BoundedLinearOperators ((G . (G,pt)),F))))
K7((BoundedLinearOperators ((G . (G,pt)),F)),(BoundedLinearOperators ((G . (G,pt)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,pt)),F)),(BoundedLinearOperators ((G . (G,pt)),F))),(BoundedLinearOperators ((G . (G,pt)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,pt)),F)),(BoundedLinearOperators ((G . (G,pt)),F))),(BoundedLinearOperators ((G . (G,pt)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,pt)),F)),(BoundedLinearOperators ((G . (G,pt)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,pt)),F))) -defined BoundedLinearOperators ((G . (G,pt)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,pt)),F))),(BoundedLinearOperators ((G . (G,pt)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,pt)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,pt)),F))),(BoundedLinearOperators ((G . (G,pt)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,pt)),F))),(BoundedLinearOperators ((G . (G,pt)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,pt)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,pt)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,pt)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,pt)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,pt)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,pt)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,pt)),F)),(BoundedLinearOperators ((G . (G,pt)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,pt)),F)),(BoundedLinearOperators ((G . (G,pt)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,pt)),F)),(BoundedLinearOperators ((G . (G,pt)),F))),(BoundedLinearOperatorsNorm ((G . (G,pt)),F)) #) is strict NORMSTR
(G,F,pt,f,m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,pt)),F))
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,pt)),F)) is non empty set
(G,(G,pt),m) is non empty Relation-like the carrier of (G . (G,pt)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,pt)), the carrier of (product G)))
K7( the carrier of (G . (G,pt)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,pt)), the carrier of (product G))) is set
f * (G,(G,pt),m) is Relation-like the carrier of (G . (G,pt)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,pt)), the carrier of F))
(G,(G,pt)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,pt)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,pt))))
K7( the carrier of (product G), the carrier of (G . (G,pt))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,pt)))) is set
(G,(G,pt)) . m is Element of the carrier of (G . (G,pt))
diff ((f * (G,(G,pt),m)),((G,(G,pt)) . m)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,pt)),F))
(G,P0) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . (G,P0) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,P0)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,P0)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,P0)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,P0)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,P0)),F))
the carrier of (G . (G,P0)) is non empty non trivial set
K266( the carrier of (G . (G,P0)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,P0)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,P0)), the carrier of F
FuncZero ( the carrier of (G . (G,P0)),F) is Relation-like the carrier of (G . (G,P0)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,P0)), the carrier of F)
the carrier of (G . (G,P0)) --> (0. F) is non empty Relation-like the carrier of (G . (G,P0)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,P0)), the carrier of F))
K7( the carrier of (G . (G,P0)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,P0)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,P0)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,P0)), the carrier of F)),(Funcs ( the carrier of (G . (G,P0)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,P0)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,P0)), the carrier of F)),(Funcs ( the carrier of (G . (G,P0)), the carrier of F))),(Funcs ( the carrier of (G . (G,P0)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,P0)), the carrier of F)),(Funcs ( the carrier of (G . (G,P0)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,P0)), the carrier of F)),(Funcs ( the carrier of (G . (G,P0)), the carrier of F))),(Funcs ( the carrier of (G . (G,P0)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,P0)), the carrier of F)),(Funcs ( the carrier of (G . (G,P0)), the carrier of F))),(Funcs ( the carrier of (G . (G,P0)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,P0)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,P0)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,P0)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,P0)), the carrier of F))),(Funcs ( the carrier of (G . (G,P0)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,P0)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,P0)), the carrier of F))),(Funcs ( the carrier of (G . (G,P0)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,P0)), the carrier of F))),(Funcs ( the carrier of (G . (G,P0)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,P0)), the carrier of F)),(FuncZero ( the carrier of (G . (G,P0)),F)),(FuncAdd ( the carrier of (G . (G,P0)),F)),(FuncExtMult ( the carrier of (G . (G,P0)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,P0)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,P0)),F)) is set
K215(K266( the carrier of (G . (G,P0)),F),(LinearOperators ((G . (G,P0)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,P0)),F)
K213(K266( the carrier of (G . (G,P0)),F),(LinearOperators ((G . (G,P0)),F))) is Relation-like K7((LinearOperators ((G . (G,P0)),F)),(LinearOperators ((G . (G,P0)),F))) -defined LinearOperators ((G . (G,P0)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,P0)),F)),(LinearOperators ((G . (G,P0)),F))),(LinearOperators ((G . (G,P0)),F))))
K7((LinearOperators ((G . (G,P0)),F)),(LinearOperators ((G . (G,P0)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,P0)),F)),(LinearOperators ((G . (G,P0)),F))),(LinearOperators ((G . (G,P0)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,P0)),F)),(LinearOperators ((G . (G,P0)),F))),(LinearOperators ((G . (G,P0)),F)))) is set
K214(K266( the carrier of (G . (G,P0)),F),(LinearOperators ((G . (G,P0)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,P0)),F))) -defined LinearOperators ((G . (G,P0)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,P0)),F))),(LinearOperators ((G . (G,P0)),F))))
K7(REAL,(LinearOperators ((G . (G,P0)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,P0)),F))),(LinearOperators ((G . (G,P0)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,P0)),F))),(LinearOperators ((G . (G,P0)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,P0)),F)),K215(K266( the carrier of (G . (G,P0)),F),(LinearOperators ((G . (G,P0)),F))),K213(K266( the carrier of (G . (G,P0)),F),(LinearOperators ((G . (G,P0)),F))),K214(K266( the carrier of (G . (G,P0)),F),(LinearOperators ((G . (G,P0)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,P0)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,P0)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,P0)),F)),(BoundedLinearOperators ((G . (G,P0)),F))) is Element of BoundedLinearOperators ((G . (G,P0)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,P0)),F)),(BoundedLinearOperators ((G . (G,P0)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,P0)),F)),(BoundedLinearOperators ((G . (G,P0)),F))) -defined BoundedLinearOperators ((G . (G,P0)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,P0)),F)),(BoundedLinearOperators ((G . (G,P0)),F))),(BoundedLinearOperators ((G . (G,P0)),F))))
K7((BoundedLinearOperators ((G . (G,P0)),F)),(BoundedLinearOperators ((G . (G,P0)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,P0)),F)),(BoundedLinearOperators ((G . (G,P0)),F))),(BoundedLinearOperators ((G . (G,P0)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,P0)),F)),(BoundedLinearOperators ((G . (G,P0)),F))),(BoundedLinearOperators ((G . (G,P0)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,P0)),F)),(BoundedLinearOperators ((G . (G,P0)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,P0)),F))) -defined BoundedLinearOperators ((G . (G,P0)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,P0)),F))),(BoundedLinearOperators ((G . (G,P0)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,P0)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,P0)),F))),(BoundedLinearOperators ((G . (G,P0)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,P0)),F))),(BoundedLinearOperators ((G . (G,P0)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,P0)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,P0)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,P0)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,P0)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,P0)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,P0)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,P0)),F)),(BoundedLinearOperators ((G . (G,P0)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,P0)),F)),(BoundedLinearOperators ((G . (G,P0)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,P0)),F)),(BoundedLinearOperators ((G . (G,P0)),F))),(BoundedLinearOperatorsNorm ((G . (G,P0)),F)) #) is strict NORMSTR
(G,F,P0,f,P1) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F))
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F)) is non empty set
(G,(G,P0),P1) is non empty Relation-like the carrier of (G . (G,P0)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,P0)), the carrier of (product G)))
K7( the carrier of (G . (G,P0)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,P0)), the carrier of (product G))) is set
f * (G,(G,P0),P1) is Relation-like the carrier of (G . (G,P0)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,P0)), the carrier of F))
(G,(G,P0)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,P0)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,P0))))
K7( the carrier of (product G), the carrier of (G . (G,P0))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,P0)))) is set
(G,(G,P0)) . P1 is Element of the carrier of (G . (G,P0))
diff ((f * (G,(G,P0),P1)),((G,(G,P0)) . P1)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F))
(G,F,P0,f,m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F))
(G,(G,P0),m) is non empty Relation-like the carrier of (G . (G,P0)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,P0)), the carrier of (product G)))
f * (G,(G,P0),m) is Relation-like the carrier of (G . (G,P0)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,P0)), the carrier of F))
(G,(G,P0)) . m is Element of the carrier of (G . (G,P0))
diff ((f * (G,(G,P0),m)),((G,(G,P0)) . m)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F))
(G,F,P0,f,P1) - (G,F,P0,f,m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F))
- (G,F,P0,f,m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F))
(G,F,P0,f,P1) + (- (G,F,P0,f,m)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F))
the addF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F)) is Relation-like K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F))) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F)) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F))))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F))) is Relation-like set
K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F))) is Relation-like set
K6(K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F)))) is set
K462( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F)), the addF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F)),(G,F,P0,f,P1),(- (G,F,P0,f,m))) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F))
||.((G,F,P0,f,P1) - (G,F,P0,f,m)).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F)) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F)) -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 ((G . (G,P0)),F)),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F)),REAL)) is non trivial V49() set
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F)) . ((G,F,P0,f,P1) - (G,F,P0,f,m)) is V11() real ext-real Element of REAL
P1 is V11() real ext-real Element of REAL
P0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
(G,P0) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . (G,P0) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,P0)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,P0)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,P0)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,P0)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,P0)),F))
the carrier of (G . (G,P0)) is non empty non trivial set
K266( the carrier of (G . (G,P0)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,P0)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,P0)), the carrier of F
FuncZero ( the carrier of (G . (G,P0)),F) is Relation-like the carrier of (G . (G,P0)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,P0)), the carrier of F)
the carrier of (G . (G,P0)) --> (0. F) is non empty Relation-like the carrier of (G . (G,P0)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,P0)), the carrier of F))
K7( the carrier of (G . (G,P0)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,P0)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,P0)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,P0)), the carrier of F)),(Funcs ( the carrier of (G . (G,P0)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,P0)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,P0)), the carrier of F)),(Funcs ( the carrier of (G . (G,P0)), the carrier of F))),(Funcs ( the carrier of (G . (G,P0)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,P0)), the carrier of F)),(Funcs ( the carrier of (G . (G,P0)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,P0)), the carrier of F)),(Funcs ( the carrier of (G . (G,P0)), the carrier of F))),(Funcs ( the carrier of (G . (G,P0)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,P0)), the carrier of F)),(Funcs ( the carrier of (G . (G,P0)), the carrier of F))),(Funcs ( the carrier of (G . (G,P0)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,P0)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,P0)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,P0)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,P0)), the carrier of F))),(Funcs ( the carrier of (G . (G,P0)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,P0)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,P0)), the carrier of F))),(Funcs ( the carrier of (G . (G,P0)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,P0)), the carrier of F))),(Funcs ( the carrier of (G . (G,P0)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,P0)), the carrier of F)),(FuncZero ( the carrier of (G . (G,P0)),F)),(FuncAdd ( the carrier of (G . (G,P0)),F)),(FuncExtMult ( the carrier of (G . (G,P0)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,P0)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,P0)),F)) is set
K215(K266( the carrier of (G . (G,P0)),F),(LinearOperators ((G . (G,P0)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,P0)),F)
K213(K266( the carrier of (G . (G,P0)),F),(LinearOperators ((G . (G,P0)),F))) is Relation-like K7((LinearOperators ((G . (G,P0)),F)),(LinearOperators ((G . (G,P0)),F))) -defined LinearOperators ((G . (G,P0)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,P0)),F)),(LinearOperators ((G . (G,P0)),F))),(LinearOperators ((G . (G,P0)),F))))
K7((LinearOperators ((G . (G,P0)),F)),(LinearOperators ((G . (G,P0)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,P0)),F)),(LinearOperators ((G . (G,P0)),F))),(LinearOperators ((G . (G,P0)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,P0)),F)),(LinearOperators ((G . (G,P0)),F))),(LinearOperators ((G . (G,P0)),F)))) is set
K214(K266( the carrier of (G . (G,P0)),F),(LinearOperators ((G . (G,P0)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,P0)),F))) -defined LinearOperators ((G . (G,P0)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,P0)),F))),(LinearOperators ((G . (G,P0)),F))))
K7(REAL,(LinearOperators ((G . (G,P0)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,P0)),F))),(LinearOperators ((G . (G,P0)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,P0)),F))),(LinearOperators ((G . (G,P0)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,P0)),F)),K215(K266( the carrier of (G . (G,P0)),F),(LinearOperators ((G . (G,P0)),F))),K213(K266( the carrier of (G . (G,P0)),F),(LinearOperators ((G . (G,P0)),F))),K214(K266( the carrier of (G . (G,P0)),F),(LinearOperators ((G . (G,P0)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,P0)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,P0)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,P0)),F)),(BoundedLinearOperators ((G . (G,P0)),F))) is Element of BoundedLinearOperators ((G . (G,P0)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,P0)),F)),(BoundedLinearOperators ((G . (G,P0)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,P0)),F)),(BoundedLinearOperators ((G . (G,P0)),F))) -defined BoundedLinearOperators ((G . (G,P0)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,P0)),F)),(BoundedLinearOperators ((G . (G,P0)),F))),(BoundedLinearOperators ((G . (G,P0)),F))))
K7((BoundedLinearOperators ((G . (G,P0)),F)),(BoundedLinearOperators ((G . (G,P0)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,P0)),F)),(BoundedLinearOperators ((G . (G,P0)),F))),(BoundedLinearOperators ((G . (G,P0)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,P0)),F)),(BoundedLinearOperators ((G . (G,P0)),F))),(BoundedLinearOperators ((G . (G,P0)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,P0)),F)),(BoundedLinearOperators ((G . (G,P0)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,P0)),F))) -defined BoundedLinearOperators ((G . (G,P0)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,P0)),F))),(BoundedLinearOperators ((G . (G,P0)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,P0)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,P0)),F))),(BoundedLinearOperators ((G . (G,P0)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,P0)),F))),(BoundedLinearOperators ((G . (G,P0)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,P0)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,P0)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,P0)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,P0)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,P0)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,P0)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,P0)),F)),(BoundedLinearOperators ((G . (G,P0)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,P0)),F)),(BoundedLinearOperators ((G . (G,P0)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,P0)),F)),(BoundedLinearOperators ((G . (G,P0)),F))),(BoundedLinearOperatorsNorm ((G . (G,P0)),F)) #) is strict NORMSTR
(G,F,P0,f,m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F))
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F)) is non empty set
(G,(G,P0),m) is non empty Relation-like the carrier of (G . (G,P0)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,P0)), the carrier of (product G)))
K7( the carrier of (G . (G,P0)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,P0)), the carrier of (product G))) is set
f * (G,(G,P0),m) is Relation-like the carrier of (G . (G,P0)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,P0)), the carrier of F))
(G,(G,P0)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,P0)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,P0))))
K7( the carrier of (product G), the carrier of (G . (G,P0))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,P0)))) is set
(G,(G,P0)) . m is Element of the carrier of (G . (G,P0))
diff ((f * (G,(G,P0),m)),((G,(G,P0)) . m)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,P0)),F))
P0 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.P0.|| is V11() real ext-real non negative Element of REAL
the normF of (product G) . P0 is V11() real ext-real Element of REAL
||.P0.|| " is V11() real ext-real non negative Element of REAL
a /. P0 is Element of the carrier of F
a . P0 is Element of the carrier of F
||.(a /. P0).|| is V11() real ext-real non negative Element of REAL
the normF of F is non empty Relation-like the carrier of F -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of F,REAL))
K7( the carrier of F,REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of F,REAL)) is non trivial V49() set
the normF of F . (a /. P0) is V11() real ext-real Element of REAL
(||.P0.|| ") * ||.(a /. P0).|| is V11() real ext-real non negative Element of REAL
(len G) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
m + P0 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),m,P0) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f /. (m + P0) is Element of the carrier of F
(f /. (m + P0)) - (f /. m) is Element of the carrier of F
- (f /. m) is Element of the carrier of F
(f /. (m + P0)) + (- (f /. m)) is Element of the carrier of F
the addF of F is Relation-like K7( the carrier of F, the carrier of F) -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of F, the carrier of F), the carrier of F))
K7( the carrier of F, the carrier of F) is Relation-like set
K7(K7( the carrier of F, the carrier of F), the carrier of F) is Relation-like set
K6(K7(K7( the carrier of F, the carrier of F), the carrier of F)) is set
K462( the carrier of F, the addF of F,(f /. (m + P0)),(- (f /. m))) is Element of the carrier of F
y0 is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
Z is Relation-like NAT -defined Function-like carr G -compatible Element of product (carr G)
pt is Relation-like NAT -defined the carrier of (product G) -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of (product G)
len pt is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
g is Relation-like NAT -defined the carrier of F -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of F
len g is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
dom pt is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
dom g is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
Sum g is Element of the carrier of F
y1 . P0 is Element of the carrier of F
w is Relation-like NAT -defined the carrier of F -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of F
dom w is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
Sum w is Element of the carrier of F
idseq (len G) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable set
id (Seg (len G)) is non empty Relation-like Seg (len G) -defined Seg (len G) -valued RAT -valued INT -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing V49() countable V221() V223() V224() V228() Element of K6(K7((Seg (len G)),(Seg (len G))))
K7((Seg (len G)),(Seg (len G))) is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued V49() countable set
K6(K7((Seg (len G)),(Seg (len G)))) is V49() V53() countable set
dom (idseq (len G)) is V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
rng (idseq (len G)) is V49() countable set
Rev (idseq (len G)) is Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable set
dom (Rev (idseq (len G))) is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
rng (Rev (idseq (len G))) is V49() countable set
Ri is non empty Relation-like Seg (len G) -defined Seg (len G) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued natural-valued V49() countable Element of K6(K7((Seg (len G)),(Seg (len G))))
K7((dom w),(dom w)) is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued V49() countable set
K6(K7((dom w),(dom w))) is V49() V53() countable set
len (idseq (len G)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
len w is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
Ri is Relation-like dom w -defined dom w -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued natural-valued V49() countable Element of K6(K7((dom w),(dom w)))
w * Ri is Relation-like dom w -defined the carrier of F -valued Function-like V49() countable Element of K6(K7((dom w), the carrier of F))
K7((dom w), the carrier of F) is Relation-like set
K6(K7((dom w), the carrier of F)) is set
dom (w * Ri) is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6((dom w))
K6((dom w)) is V49() V53() countable set
dom Ri is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6((dom w))
Rev w is Relation-like NAT -defined the carrier of F -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of F
dom (Rev w) is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
gw is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
(len G) - gw is V11() real ext-real Element of REAL
K38(gw) is V11() real ext-real non positive set
K36((len G),K38(gw)) is V11() real ext-real set
yseq is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
yseq is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
yseq + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
((len G) - gw) + 1 is V11() real ext-real Element of REAL
1 - 1 is V11() real ext-real V132() V160() Element of INT
K38(1) is non empty V11() real ext-real non positive negative set
K36(1,K38(1)) is V11() real ext-real set
gw - 1 is V11() real ext-real Element of REAL
K36(gw,K38(1)) is V11() real ext-real set
(len G) - (gw - 1) is V11() real ext-real Element of REAL
K38((gw - 1)) is V11() real ext-real set
K36((len G),K38((gw - 1))) is V11() real ext-real set
yseq + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
(Rev w) . gw is set
(len (idseq (len G))) - gw is V11() real ext-real Element of REAL
K36((len (idseq (len G))),K38(gw)) is V11() real ext-real set
((len (idseq (len G))) - gw) + 1 is V11() real ext-real Element of REAL
w . (((len (idseq (len G))) - gw) + 1) is set
(idseq (len G)) . (((len (idseq (len G))) - gw) + 1) is set
w . ((idseq (len G)) . (((len (idseq (len G))) - gw) + 1)) is set
(Rev (idseq (len G))) . gw is set
w . ((Rev (idseq (len G))) . gw) is set
wRi is Relation-like NAT -defined the carrier of F -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of F
wRi . gw is set
Sum (Rev w) is Element of the carrier of F
gw is Relation-like NAT -defined the carrier of F -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of F
len gw is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
dom gw is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
yseq is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
gw . yseq is set
g /. yseq is Element of the carrier of F
(Rev w) /. yseq is Element of the carrier of F
(g /. yseq) - ((Rev w) /. yseq) is Element of the carrier of F
- ((Rev w) /. yseq) is Element of the carrier of F
(g /. yseq) + (- ((Rev w) /. yseq)) is Element of the carrier of F
K462( the carrier of F, the addF of F,(g /. yseq),(- ((Rev w) /. yseq))) is Element of the carrier of F
len (Rev w) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
Sum gw is Element of the carrier of F
(Sum g) - (Sum (Rev w)) is Element of the carrier of F
- (Sum (Rev w)) is Element of the carrier of F
(Sum g) + (- (Sum (Rev w))) is Element of the carrier of F
K462( the carrier of F, the addF of F,(Sum g),(- (Sum (Rev w)))) is Element of the carrier of F
||.P0.|| * ((t / 2) / (len G)) is V11() real ext-real Element of REAL
yseq is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
gw /. yseq is Element of the carrier of F
||.(gw /. yseq).|| is V11() real ext-real non negative Element of REAL
the normF of F . (gw /. yseq) is V11() real ext-real Element of REAL
g /. yseq is Element of the carrier of F
pt /. yseq is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
m + (pt /. yseq) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),m,(pt /. yseq)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f /. (m + (pt /. yseq)) is Element of the carrier of F
yseq + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
pt /. (yseq + 1) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
m + (pt /. (yseq + 1)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),m,(pt /. (yseq + 1))) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f /. (m + (pt /. (yseq + 1))) is Element of the carrier of F
(f /. (m + (pt /. yseq))) - (f /. (m + (pt /. (yseq + 1)))) is Element of the carrier of F
- (f /. (m + (pt /. (yseq + 1)))) is Element of the carrier of F
(f /. (m + (pt /. yseq))) + (- (f /. (m + (pt /. (yseq + 1))))) is Element of the carrier of F
K462( the carrier of F, the addF of F,(f /. (m + (pt /. yseq))),(- (f /. (m + (pt /. (yseq + 1)))))) is Element of the carrier of F
(len G) + yseq is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
((len G) + 1) - yseq is V11() real ext-real V132() V160() Element of INT
K38(yseq) is V11() real ext-real non positive set
K36(((len G) + 1),K38(yseq)) is V11() real ext-real set
((len G) + 1) -' yseq is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
yseq is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
w /. (((len G) + 1) -' yseq) is Element of the carrier of F
w . (((len G) + 1) -' yseq) is set
(G,(((len G) + 1) -' yseq)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . (G,(((len G) + 1) -' yseq)) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
(G,F,(((len G) + 1) -' yseq),f,m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))
R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)))
R_VectorSpace_of_LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,(((len G) + 1) -' yseq))),F))
the carrier of (G . (G,(((len G) + 1) -' yseq))) is non empty non trivial set
K266( the carrier of (G . (G,(((len G) + 1) -' yseq))),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F
FuncZero ( the carrier of (G . (G,(((len G) + 1) -' yseq))),F) is Relation-like the carrier of (G . (G,(((len G) + 1) -' yseq))) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F)
the carrier of (G . (G,(((len G) + 1) -' yseq))) --> (0. F) is non empty Relation-like the carrier of (G . (G,(((len G) + 1) -' yseq))) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F))
K7( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,(((len G) + 1) -' yseq))),F) is Relation-like K7((Funcs ( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F)),(Funcs ( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F))) -defined Funcs ( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F)),(Funcs ( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F))),(Funcs ( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F)),(Funcs ( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F)),(Funcs ( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F))),(Funcs ( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F)),(Funcs ( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F))),(Funcs ( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,(((len G) + 1) -' yseq))),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F))) -defined Funcs ( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F))),(Funcs ( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F))),(Funcs ( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F))),(Funcs ( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F)),(FuncZero ( the carrier of (G . (G,(((len G) + 1) -' yseq))),F)),(FuncAdd ( the carrier of (G . (G,(((len G) + 1) -' yseq))),F)),(FuncExtMult ( the carrier of (G . (G,(((len G) + 1) -' yseq))),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,(((len G) + 1) -' yseq))),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,(((len G) + 1) -' yseq))),F)) is set
K215(K266( the carrier of (G . (G,(((len G) + 1) -' yseq))),F),(LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)
K213(K266( the carrier of (G . (G,(((len G) + 1) -' yseq))),F),(LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))) is Relation-like K7((LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)),(LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))) -defined LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)),(LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))),(LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))))
K7((LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)),(LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)),(LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))),(LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)),(LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))),(LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)))) is set
K214(K266( the carrier of (G . (G,(((len G) + 1) -' yseq))),F),(LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))) -defined LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))),(LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))))
K7(REAL,(LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))),(LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))),(LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)),K215(K266( the carrier of (G . (G,(((len G) + 1) -' yseq))),F),(LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))),K213(K266( the carrier of (G . (G,(((len G) + 1) -' yseq))),F),(LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))),K214(K266( the carrier of (G . (G,(((len G) + 1) -' yseq))),F),(LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)),(BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))) is Element of BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)),(BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)),(BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))) -defined BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)),(BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))),(BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))))
K7((BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)),(BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)),(BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))),(BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)),(BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))),(BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)),(BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))) -defined BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))),(BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))),(BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))),(BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,(((len G) + 1) -' yseq))),F) is non empty Relation-like BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)),REAL))
K7((BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)),(BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)),(BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)),(BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))),(BoundedLinearOperatorsNorm ((G . (G,(((len G) + 1) -' yseq))),F)) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)) is non empty set
(G,(G,(((len G) + 1) -' yseq)),m) is non empty Relation-like the carrier of (G . (G,(((len G) + 1) -' yseq))) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of (product G)))
K7( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of (product G))) is set
f * (G,(G,(((len G) + 1) -' yseq)),m) is Relation-like the carrier of (G . (G,(((len G) + 1) -' yseq))) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F))
(G,(G,(((len G) + 1) -' yseq))) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,(((len G) + 1) -' yseq))) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,(((len G) + 1) -' yseq)))))
K7( the carrier of (product G), the carrier of (G . (G,(((len G) + 1) -' yseq)))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,(((len G) + 1) -' yseq))))) is set
(G,(G,(((len G) + 1) -' yseq))) . m is Element of the carrier of (G . (G,(((len G) + 1) -' yseq)))
diff ((f * (G,(G,(((len G) + 1) -' yseq)),m)),((G,(G,(((len G) + 1) -' yseq))) . m)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))
(G,(G,(((len G) + 1) -' yseq))) . P0 is Element of the carrier of (G . (G,(((len G) + 1) -' yseq)))
(G,F,(((len G) + 1) -' yseq),f,m) . ((G,(G,(((len G) + 1) -' yseq))) . P0) is Element of the carrier of F
yseq is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
pt /. yseq is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
j is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.j.|| is V11() real ext-real non negative Element of REAL
the normF of (product G) . j is V11() real ext-real Element of REAL
m + (pt /. yseq) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),m,(pt /. yseq)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(m + (pt /. yseq)) - m is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(m + (pt /. yseq)) + (- m) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),(m + (pt /. yseq)),(- m)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
m - m is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
m + (- m) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),m,(- m)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(pt /. yseq) + (m - m) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),(pt /. yseq),(m - m)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(pt /. yseq) + (0. (product G)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),(pt /. yseq),(0. (product G))) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.((m + (pt /. yseq)) - m).|| is V11() real ext-real non negative Element of REAL
the normF of (product G) . ((m + (pt /. yseq)) - m) is V11() real ext-real Element of REAL
Seg ((len G) + 1) is non empty V49() (len G) + 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= (len G) + 1 ) } is set
(m + (pt /. yseq)) - m is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(m + (pt /. yseq)) + (- m) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),(m + (pt /. yseq)),(- m)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.((m + (pt /. yseq)) - m).|| is V11() real ext-real non negative Element of REAL
the normF of (product G) . ((m + (pt /. yseq)) - m) is V11() real ext-real Element of REAL
(m + (pt /. (yseq + 1))) - m is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(m + (pt /. (yseq + 1))) + (- m) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),(m + (pt /. (yseq + 1))),(- m)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.((m + (pt /. (yseq + 1))) - m).|| is V11() real ext-real non negative Element of REAL
the normF of (product G) . ((m + (pt /. (yseq + 1))) - m) is V11() real ext-real Element of REAL
(G,(G,(((len G) + 1) -' yseq)),(m + (pt /. (yseq + 1)))) is non empty Relation-like the carrier of (G . (G,(((len G) + 1) -' yseq))) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of (product G)))
(G,(G,(((len G) + 1) -' yseq))) . (m + P0) is Element of the carrier of (G . (G,(((len G) + 1) -' yseq)))
(G,(G,(((len G) + 1) -' yseq)),(m + (pt /. (yseq + 1)))) . ((G,(G,(((len G) + 1) -' yseq))) . (m + P0)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,(G,(((len G) + 1) -' yseq))) . (m + (pt /. (yseq + 1))) is Element of the carrier of (G . (G,(((len G) + 1) -' yseq)))
((G,(G,(((len G) + 1) -' yseq))) . (m + P0)) - ((G,(G,(((len G) + 1) -' yseq))) . (m + (pt /. (yseq + 1)))) is Element of the carrier of (G . (G,(((len G) + 1) -' yseq)))
- ((G,(G,(((len G) + 1) -' yseq))) . (m + (pt /. (yseq + 1)))) is Element of the carrier of (G . (G,(((len G) + 1) -' yseq)))
((G,(G,(((len G) + 1) -' yseq))) . (m + P0)) + (- ((G,(G,(((len G) + 1) -' yseq))) . (m + (pt /. (yseq + 1))))) is Element of the carrier of (G . (G,(((len G) + 1) -' yseq)))
the addF of (G . (G,(((len G) + 1) -' yseq))) is Relation-like K7( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of (G . (G,(((len G) + 1) -' yseq)))) -defined the carrier of (G . (G,(((len G) + 1) -' yseq))) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of (G . (G,(((len G) + 1) -' yseq)))), the carrier of (G . (G,(((len G) + 1) -' yseq)))))
K7( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of (G . (G,(((len G) + 1) -' yseq)))) is Relation-like set
K7(K7( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of (G . (G,(((len G) + 1) -' yseq)))), the carrier of (G . (G,(((len G) + 1) -' yseq)))) is Relation-like set
K6(K7(K7( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of (G . (G,(((len G) + 1) -' yseq)))), the carrier of (G . (G,(((len G) + 1) -' yseq))))) is set
K462( the carrier of (G . (G,(((len G) + 1) -' yseq))), the addF of (G . (G,(((len G) + 1) -' yseq))),((G,(G,(((len G) + 1) -' yseq))) . (m + P0)),(- ((G,(G,(((len G) + 1) -' yseq))) . (m + (pt /. (yseq + 1)))))) is Element of the carrier of (G . (G,(((len G) + 1) -' yseq)))
yseq is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
yseq - m is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
yseq + (- m) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),yseq,(- m)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.(yseq - m).|| is V11() real ext-real non negative Element of REAL
the normF of (product G) . (yseq - m) is V11() real ext-real Element of REAL
(G,F,(((len G) + 1) -' yseq),f,yseq) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))
(G,(G,(((len G) + 1) -' yseq)),yseq) is non empty Relation-like the carrier of (G . (G,(((len G) + 1) -' yseq))) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of (product G)))
f * (G,(G,(((len G) + 1) -' yseq)),yseq) is Relation-like the carrier of (G . (G,(((len G) + 1) -' yseq))) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,(((len G) + 1) -' yseq))), the carrier of F))
(G,(G,(((len G) + 1) -' yseq))) . yseq is Element of the carrier of (G . (G,(((len G) + 1) -' yseq)))
diff ((f * (G,(G,(((len G) + 1) -' yseq)),yseq)),((G,(G,(((len G) + 1) -' yseq))) . yseq)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))
(G,F,(((len G) + 1) -' yseq),f,yseq) - (G,F,(((len G) + 1) -' yseq),f,m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))
- (G,F,(((len G) + 1) -' yseq),f,m) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))
(G,F,(((len G) + 1) -' yseq),f,yseq) + (- (G,F,(((len G) + 1) -' yseq),f,m)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))
the addF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)) is Relation-like K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))) is Relation-like set
K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))) is Relation-like set
K6(K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)))) is set
K462( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)), the addF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)),(G,F,(((len G) + 1) -' yseq),f,yseq),(- (G,F,(((len G) + 1) -' yseq),f,m))) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F))
||.((G,F,(((len G) + 1) -' yseq),f,yseq) - (G,F,(((len G) + 1) -' yseq),f,m)).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)) -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 ((G . (G,(((len G) + 1) -' yseq))),F)),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)),REAL)) is non trivial V49() set
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,(((len G) + 1) -' yseq))),F)) . ((G,F,(((len G) + 1) -' yseq),f,yseq) - (G,F,(((len G) + 1) -' yseq),f,m)) is V11() real ext-real Element of REAL
((f /. (m + (pt /. yseq))) - (f /. (m + (pt /. (yseq + 1))))) - ((G,F,(((len G) + 1) -' yseq),f,m) . ((G,(G,(((len G) + 1) -' yseq))) . P0)) is Element of the carrier of F
- ((G,F,(((len G) + 1) -' yseq),f,m) . ((G,(G,(((len G) + 1) -' yseq))) . P0)) is Element of the carrier of F
((f /. (m + (pt /. yseq))) - (f /. (m + (pt /. (yseq + 1))))) + (- ((G,F,(((len G) + 1) -' yseq),f,m) . ((G,(G,(((len G) + 1) -' yseq))) . P0))) is Element of the carrier of F
K462( the carrier of F, the addF of F,((f /. (m + (pt /. yseq))) - (f /. (m + (pt /. (yseq + 1))))),(- ((G,F,(((len G) + 1) -' yseq),f,m) . ((G,(G,(((len G) + 1) -' yseq))) . P0)))) is Element of the carrier of F
||.(((f /. (m + (pt /. yseq))) - (f /. (m + (pt /. (yseq + 1))))) - ((G,F,(((len G) + 1) -' yseq),f,m) . ((G,(G,(((len G) + 1) -' yseq))) . P0))).|| is V11() real ext-real non negative Element of REAL
the normF of F . (((f /. (m + (pt /. yseq))) - (f /. (m + (pt /. (yseq + 1))))) - ((G,F,(((len G) + 1) -' yseq),f,m) . ((G,(G,(((len G) + 1) -' yseq))) . P0))) is V11() real ext-real Element of REAL
||.((G,(G,(((len G) + 1) -' yseq))) . P0).|| is V11() real ext-real non negative Element of REAL
the normF of (G . (G,(((len G) + 1) -' yseq))) is non empty Relation-like the carrier of (G . (G,(((len G) + 1) -' yseq))) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (G . (G,(((len G) + 1) -' yseq))),REAL))
K7( the carrier of (G . (G,(((len G) + 1) -' yseq))),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (G . (G,(((len G) + 1) -' yseq))),REAL)) is non trivial V49() set
the normF of (G . (G,(((len G) + 1) -' yseq))) . ((G,(G,(((len G) + 1) -' yseq))) . P0) is V11() real ext-real Element of REAL
||.((G,(G,(((len G) + 1) -' yseq))) . P0).|| * ((t / 2) / (len G)) is V11() real ext-real Element of REAL
Seg (len (Rev w)) is V49() len (Rev w) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b₁ & b₁ <= len (Rev w) ) } is set
(Rev w) /. yseq is Element of the carrier of F
(Rev w) . yseq is set
(len G) - yseq is V11() real ext-real V132() V160() Element of INT
K36((len G),K38(yseq)) is V11() real ext-real set
((len G) - yseq) + 1 is V11() real ext-real V132() V160() Element of INT
w . (((len G) - yseq) + 1) is set
gw . yseq is set
yseq is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
gw /. yseq is Element of the carrier of F
||.(gw /. yseq).|| is V11() real ext-real non negative Element of REAL
the normF of F . (gw /. yseq) is V11() real ext-real Element of REAL
yseq is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
dom yseq is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
len yseq is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
yseq is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
yseq . yseq is V11() real ext-real Element of REAL
gw /. yseq is Element of the carrier of F
||.(gw /. yseq).|| is V11() real ext-real non negative Element of REAL
the normF of F . (gw /. yseq) is V11() real ext-real Element of REAL
REAL (len G) is non empty functional FinSequence-membered FinSequenceSet of REAL
(len G) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
REAL * is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like V49() FinSequence-like FinSubsequence-like countable Element of REAL * : len b₁ = len G } is set
||.(Sum gw).|| is V11() real ext-real non negative Element of REAL
the normF of F . (Sum gw) is V11() real ext-real Element of REAL
yseq is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() len G -element FinSequence-like FinSubsequence-like countable Element of REAL (len G)
Sum yseq is V11() real ext-real Element of REAL
(len G) |-> (||.P0.|| * ((t / 2) / (len G))) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() len G -element FinSequence-like FinSubsequence-like countable Element of (len G) -tuples_on REAL
(Seg (len G)) --> (||.P0.|| * ((t / 2) / (len G))) is non empty Relation-like NAT -defined Seg (len G) -defined Seg (len G) -defined REAL -valued {(||.P0.|| * ((t / 2) / (len G)))} -valued Function-like constant total total quasi_total complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable Element of K6(K7((Seg (len G)),{(||.P0.|| * ((t / 2) / (len G)))}))
{(||.P0.|| * ((t / 2) / (len G)))} is non empty trivial V49() 1 -element complex-membered ext-real-membered real-membered countable left_end right_end bounded_below bounded_above real-bounded set
K7((Seg (len G)),{(||.P0.|| * ((t / 2) / (len G)))}) is Relation-like complex-valued ext-real-valued real-valued V49() countable set
K6(K7((Seg (len G)),{(||.P0.|| * ((t / 2) / (len G)))})) is V49() V53() countable set
j is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
yseq . j is V11() real ext-real Element of REAL
((len G) |-> (||.P0.|| * ((t / 2) / (len G)))) . j is V11() real ext-real Element of REAL
gw /. j is Element of the carrier of F
||.(gw /. j).|| is V11() real ext-real non negative Element of REAL
the normF of F . (gw /. j) is V11() real ext-real Element of REAL
Sum ((len G) |-> (||.P0.|| * ((t / 2) / (len G)))) is V11() real ext-real Element of REAL
(len G) * (||.P0.|| * ((t / 2) / (len G))) is V11() real ext-real Element of REAL
||.(a /. P0).|| * (||.P0.|| ") is V11() real ext-real non negative Element of REAL
(len G) * ||.P0.|| is V11() real ext-real non negative Element of REAL
((len G) * ||.P0.||) * ((t / 2) / (len G)) is V11() real ext-real Element of REAL
(((len G) * ||.P0.||) * ((t / 2) / (len G))) * (||.P0.|| ") is V11() real ext-real Element of REAL
((t / 2) / (len G)) * ||.P0.|| is V11() real ext-real Element of REAL
(((t / 2) / (len G)) * ||.P0.||) * (||.P0.|| ") is V11() real ext-real Element of REAL
(len G) * ((((t / 2) / (len G)) * ||.P0.||) * (||.P0.|| ")) is V11() real ext-real Element of REAL
(len G) * ((t / 2) / (len G)) is V11() real ext-real Element of REAL
tm is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
t is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like RestFunc-like Element of K6(K7( the carrier of (product G), the carrier of F))
Sw1 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f /. Sw1 is Element of the carrier of F
(f /. Sw1) - (f /. m) is Element of the carrier of F
- (f /. m) is Element of the carrier of F
(f /. Sw1) + (- (f /. m)) is Element of the carrier of F
the addF of F is Relation-like K7( the carrier of F, the carrier of F) -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of F, the carrier of F), the carrier of F))
K7( the carrier of F, the carrier of F) is Relation-like set
K7(K7( the carrier of F, the carrier of F), the carrier of F) is Relation-like set
K6(K7(K7( the carrier of F, the carrier of F), the carrier of F)) is set
K462( the carrier of F, the addF of F,(f /. Sw1),(- (f /. m))) is Element of the carrier of F
Sw1 - m is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
Sw1 + (- m) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),Sw1,(- m)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
tm . (Sw1 - m) is Element of the carrier of F
t /. (Sw1 - m) is Element of the carrier of F
(tm . (Sw1 - m)) + (t /. (Sw1 - m)) is Element of the carrier of F
K462( the carrier of F, the addF of F,(tm . (Sw1 - m)),(t /. (Sw1 - m))) is Element of the carrier of F
dom t is Element of K6( the carrier of (product G))
t . (Sw1 - m) is set
m + (Sw1 - m) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),m,(Sw1 - m)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f /. (m + (Sw1 - m)) is Element of the carrier of F
(f /. (m + (Sw1 - m))) - (f /. m) is Element of the carrier of F
(f /. (m + (Sw1 - m))) + (- (f /. m)) is Element of the carrier of F
K462( the carrier of F, the addF of F,(f /. (m + (Sw1 - m))),(- (f /. m))) is Element of the carrier of F
((f /. (m + (Sw1 - m))) - (f /. m)) - (tm . (Sw1 - m)) is Element of the carrier of F
- (tm . (Sw1 - m)) is Element of the carrier of F
((f /. (m + (Sw1 - m))) - (f /. m)) + (- (tm . (Sw1 - m))) is Element of the carrier of F
K462( the carrier of F, the addF of F,((f /. (m + (Sw1 - m))) - (f /. m)),(- (tm . (Sw1 - m)))) is Element of the carrier of F
(tm . (Sw1 - m)) - (tm . (Sw1 - m)) is Element of the carrier of F
(tm . (Sw1 - m)) + (- (tm . (Sw1 - m))) is Element of the carrier of F
K462( the carrier of F, the addF of F,(tm . (Sw1 - m)),(- (tm . (Sw1 - m)))) is Element of the carrier of F
((f /. (m + (Sw1 - m))) - (f /. m)) - ((tm . (Sw1 - m)) - (tm . (Sw1 - m))) is Element of the carrier of F
- ((tm . (Sw1 - m)) - (tm . (Sw1 - m))) is Element of the carrier of F
((f /. (m + (Sw1 - m))) - (f /. m)) + (- ((tm . (Sw1 - m)) - (tm . (Sw1 - m)))) is Element of the carrier of F
K462( the carrier of F, the addF of F,((f /. (m + (Sw1 - m))) - (f /. m)),(- ((tm . (Sw1 - m)) - (tm . (Sw1 - m))))) is Element of the carrier of F
((f /. (m + (Sw1 - m))) - (f /. m)) - (0. F) is Element of the carrier of F
- (0. F) is Element of the carrier of F
((f /. (m + (Sw1 - m))) - (f /. m)) + (- (0. F)) is Element of the carrier of F
K462( the carrier of F, the addF of F,((f /. (m + (Sw1 - m))) - (f /. m)),(- (0. F))) is Element of the carrier of F
m - m is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
m + (- m) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),m,(- m)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
Sw1 - (m - m) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
- (m - m) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
Sw1 + (- (m - m)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),Sw1,(- (m - m))) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f /. (Sw1 - (m - m)) is Element of the carrier of F
(f /. (Sw1 - (m - m))) - (f /. m) is Element of the carrier of F
(f /. (Sw1 - (m - m))) + (- (f /. m)) is Element of the carrier of F
K462( the carrier of F, the addF of F,(f /. (Sw1 - (m - m))),(- (f /. m))) is Element of the carrier of F
Sw1 - (0. (product G)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
- (0. (product G)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
Sw1 + (- (0. (product G))) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
K462( the carrier of (product G), the addF of (product G),Sw1,(- (0. (product G)))) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
f /. (Sw1 - (0. (product G))) is Element of the carrier of F
(f /. (Sw1 - (0. (product G)))) - (f /. m) is Element of the carrier of F
(f /. (Sw1 - (0. (product G)))) + (- (f /. m)) is Element of the carrier of F
K462( the carrier of F, the addF of F,(f /. (Sw1 - (0. (product G)))),(- (f /. m))) is Element of the carrier of F
Sw1 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(diff (f,m)) . Sw1 is Element of the carrier of F
G is non empty Relation-like NAT -defined Function-like V49() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding () set
product G is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like constituted-Functions constituted-FinSeqs NORMSTR
the carrier of (product G) is non empty non trivial set
K6( the carrier of (product G)) is set
dom G is non empty V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
F is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of F is non empty non trivial set
K7( the carrier of (product G), the carrier of F) is Relation-like set
K6(K7( the carrier of (product G), the carrier of F)) is set
R_NormSpace_of_BoundedLinearOperators ((product G),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((product G),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((product G),F)))
R_VectorSpace_of_LinearOperators ((product G),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((product G),F) is non empty functional Element of K6( the carrier of K266( the carrier of (product G),F))
K266( the carrier of (product G),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (product G), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (product G), the carrier of F
FuncZero ( the carrier of (product G),F) is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (product G), the carrier of F)
0. F is V68(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
the carrier of (product G) --> (0. F) is non empty Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (product G), the carrier of F))
FuncAdd ( the carrier of (product G),F) is Relation-like K7((Funcs ( the carrier of (product G), the carrier of F)),(Funcs ( the carrier of (product G), the carrier of F))) -defined Funcs ( the carrier of (product G), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (product G), the carrier of F)),(Funcs ( the carrier of (product G), the carrier of F))),(Funcs ( the carrier of (product G), the carrier of F))))
K7((Funcs ( the carrier of (product G), the carrier of F)),(Funcs ( the carrier of (product G), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (product G), the carrier of F)),(Funcs ( the carrier of (product G), the carrier of F))),(Funcs ( the carrier of (product G), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (product G), the carrier of F)),(Funcs ( the carrier of (product G), the carrier of F))),(Funcs ( the carrier of (product G), the carrier of F)))) is set
FuncExtMult ( the carrier of (product G),F) is Relation-like K7(REAL,(Funcs ( the carrier of (product G), the carrier of F))) -defined Funcs ( the carrier of (product G), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (product G), the carrier of F))),(Funcs ( the carrier of (product G), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (product G), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (product G), the carrier of F))),(Funcs ( the carrier of (product G), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (product G), the carrier of F))),(Funcs ( the carrier of (product G), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (product G), the carrier of F)),(FuncZero ( the carrier of (product G),F)),(FuncAdd ( the carrier of (product G),F)),(FuncExtMult ( the carrier of (product G),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (product G),F) is non empty set
K6( the carrier of K266( the carrier of (product G),F)) is set
K215(K266( the carrier of (product G),F),(LinearOperators ((product G),F))) is Relation-like Function-like Element of LinearOperators ((product G),F)
K213(K266( the carrier of (product G),F),(LinearOperators ((product G),F))) is Relation-like K7((LinearOperators ((product G),F)),(LinearOperators ((product G),F))) -defined LinearOperators ((product G),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((product G),F)),(LinearOperators ((product G),F))),(LinearOperators ((product G),F))))
K7((LinearOperators ((product G),F)),(LinearOperators ((product G),F))) is Relation-like set
K7(K7((LinearOperators ((product G),F)),(LinearOperators ((product G),F))),(LinearOperators ((product G),F))) is Relation-like set
K6(K7(K7((LinearOperators ((product G),F)),(LinearOperators ((product G),F))),(LinearOperators ((product G),F)))) is set
K214(K266( the carrier of (product G),F),(LinearOperators ((product G),F))) is Relation-like K7(REAL,(LinearOperators ((product G),F))) -defined LinearOperators ((product G),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((product G),F))),(LinearOperators ((product G),F))))
K7(REAL,(LinearOperators ((product G),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((product G),F))),(LinearOperators ((product G),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((product G),F))),(LinearOperators ((product G),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((product G),F)),K215(K266( the carrier of (product G),F),(LinearOperators ((product G),F))),K213(K266( the carrier of (product G),F),(LinearOperators ((product G),F))),K214(K266( the carrier of (product G),F),(LinearOperators ((product G),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((product G),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((product G),F))) is set
K215((R_VectorSpace_of_LinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))) is Element of BoundedLinearOperators ((product G),F)
K213((R_VectorSpace_of_LinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))) is Relation-like K7((BoundedLinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))) -defined BoundedLinearOperators ((product G),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))),(BoundedLinearOperators ((product G),F))))
K7((BoundedLinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))),(BoundedLinearOperators ((product G),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))),(BoundedLinearOperators ((product G),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((product G),F))) -defined BoundedLinearOperators ((product G),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((product G),F))),(BoundedLinearOperators ((product G),F))))
K7(REAL,(BoundedLinearOperators ((product G),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((product G),F))),(BoundedLinearOperators ((product G),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((product G),F))),(BoundedLinearOperators ((product G),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((product G),F) is non empty Relation-like BoundedLinearOperators ((product G),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((product G),F)),REAL))
K7((BoundedLinearOperators ((product G),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((product G),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((product G),F)),K215((R_VectorSpace_of_LinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))),K213((R_VectorSpace_of_LinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))),K214((R_VectorSpace_of_LinearOperators ((product G),F)),(BoundedLinearOperators ((product G),F))),(BoundedLinearOperatorsNorm ((product G),F)) #) is strict NORMSTR
f is Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of F))
X is Element of K6( the carrier of (product G))
f `| X is Relation-like the carrier of (product G) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)) -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))))
the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)) is non empty set
K7( the carrier of (product G), the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)))) is set
len G is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len G) is non empty V49() len G -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(NAT)
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT : ( 1 <= b<sub>1</sub> & b<sub>1</sub> <= len G ) } is set
i is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
(G,i) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . (G,i) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,i)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,i)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,i)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,i)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,i)),F))
the carrier of (G . (G,i)) is non empty non trivial set
K266( the carrier of (G . (G,i)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,i)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,i)), the carrier of F
FuncZero ( the carrier of (G . (G,i)),F) is Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,i)), the carrier of F)
the carrier of (G . (G,i)) --> (0. F) is non empty Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,i)), the carrier of F))
K7( the carrier of (G . (G,i)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,i)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,i)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,i)), the carrier of F)),(Funcs ( the carrier of (G . (G,i)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,i)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,i)), the carrier of F)),(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,i)), the carrier of F)),(Funcs ( the carrier of (G . (G,i)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,i)), the carrier of F)),(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,i)), the carrier of F)),(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,i)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,i)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,i)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,i)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,i)), the carrier of F)),(FuncZero ( the carrier of (G . (G,i)),F)),(FuncAdd ( the carrier of (G . (G,i)),F)),(FuncExtMult ( the carrier of (G . (G,i)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,i)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,i)),F)) is set
K215(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,i)),F)
K213(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))) is Relation-like K7((LinearOperators ((G . (G,i)),F)),(LinearOperators ((G . (G,i)),F))) -defined LinearOperators ((G . (G,i)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,i)),F)),(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F))))
K7((LinearOperators ((G . (G,i)),F)),(LinearOperators ((G . (G,i)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,i)),F)),(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,i)),F)),(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F)))) is set
K214(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,i)),F))) -defined LinearOperators ((G . (G,i)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F))))
K7(REAL,(LinearOperators ((G . (G,i)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,i)),F)),K215(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))),K213(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))),K214(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,i)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,i)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))) is Element of BoundedLinearOperators ((G . (G,i)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))) -defined BoundedLinearOperators ((G . (G,i)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F))))
K7((BoundedLinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,i)),F))) -defined BoundedLinearOperators ((G . (G,i)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,i)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,i)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,i)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,i)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,i)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,i)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,i)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperatorsNorm ((G . (G,i)),F)) #) is strict NORMSTR
(G,F,i,f,X) is Relation-like the carrier of (product G) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)) -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))))
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)) is non empty set
K7( the carrier of (product G), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)))) is set
dom (G,F,i,f,X) is Element of K6( the carrier of (product G))
i0 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
y0 is V11() real ext-real Element of REAL
(G,F,i,f,X) /. i0 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
dom f is Element of K6( the carrier of (product G))
i is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
dom (f `| X) is Element of K6( the carrier of (product G))
i is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(f `| X) /. i is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
i0 is V11() real ext-real Element of REAL
2 * (len G) is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable left_end right_end bounded_below bounded_above real-bounded Element of NAT
i0 / (2 * (len G)) is V11() real ext-real Element of REAL
K39((2 * (len G))) is non empty V11() real ext-real positive non negative set
K37(i0,K39((2 * (len G)))) is V11() real ext-real set
y0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
r is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
(G,r) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . (G,r) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,r)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,r)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,r)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,r)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,r)),F))
the carrier of (G . (G,r)) is non empty non trivial set
K266( the carrier of (G . (G,r)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,r)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,r)), the carrier of F
FuncZero ( the carrier of (G . (G,r)),F) is Relation-like the carrier of (G . (G,r)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,r)), the carrier of F)
the carrier of (G . (G,r)) --> (0. F) is non empty Relation-like the carrier of (G . (G,r)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,r)), the carrier of F))
K7( the carrier of (G . (G,r)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,r)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,r)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,r)), the carrier of F)),(Funcs ( the carrier of (G . (G,r)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,r)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,r)), the carrier of F)),(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,r)), the carrier of F)),(Funcs ( the carrier of (G . (G,r)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,r)), the carrier of F)),(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,r)), the carrier of F)),(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,r)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,r)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,r)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,r)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,r)), the carrier of F))),(Funcs ( the carrier of (G . (G,r)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,r)), the carrier of F)),(FuncZero ( the carrier of (G . (G,r)),F)),(FuncAdd ( the carrier of (G . (G,r)),F)),(FuncExtMult ( the carrier of (G . (G,r)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,r)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,r)),F)) is set
K215(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,r)),F)
K213(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))) is Relation-like K7((LinearOperators ((G . (G,r)),F)),(LinearOperators ((G . (G,r)),F))) -defined LinearOperators ((G . (G,r)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,r)),F)),(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F))))
K7((LinearOperators ((G . (G,r)),F)),(LinearOperators ((G . (G,r)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,r)),F)),(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,r)),F)),(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F)))) is set
K214(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,r)),F))) -defined LinearOperators ((G . (G,r)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F))))
K7(REAL,(LinearOperators ((G . (G,r)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,r)),F))),(LinearOperators ((G . (G,r)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,r)),F)),K215(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))),K213(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))),K214(K266( the carrier of (G . (G,r)),F),(LinearOperators ((G . (G,r)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,r)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,r)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))) is Element of BoundedLinearOperators ((G . (G,r)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))) -defined BoundedLinearOperators ((G . (G,r)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F))))
K7((BoundedLinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,r)),F))) -defined BoundedLinearOperators ((G . (G,r)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,r)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperators ((G . (G,r)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,r)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,r)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,r)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,r)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,r)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,r)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,r)),F)),(BoundedLinearOperators ((G . (G,r)),F))),(BoundedLinearOperatorsNorm ((G . (G,r)),F)) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F)) is non empty set
(G,F,r,f,X) is Relation-like the carrier of (product G) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F)) -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F))))
K7( the carrier of (product G), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F)))) is set
(G,F,r,f,X) /. i is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,r)),F))
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
DD is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
a is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
a - i is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
- i is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
a + (- i) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
the addF of (product G) is Relation-like K7( the carrier of (product G), the carrier of (product G)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)))
K7( the carrier of (product G), the carrier of (product G)) is Relation-like set
K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)) is Relation-like set
K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G))) is set
K462( the carrier of (product G), the addF of (product G),a,(- i)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.(a - i).|| is V11() real ext-real non negative Element of REAL
the normF of (product G) is non empty Relation-like the carrier of (product G) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (product G),REAL))
K7( the carrier of (product G),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (product G),REAL)) is non trivial V49() set
the normF of (product G) . (a - i) is V11() real ext-real Element of REAL
(G,DD) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . (G,DD) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,DD)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,DD)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,DD)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,DD)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,DD)),F))
the carrier of (G . (G,DD)) is non empty non trivial set
K266( the carrier of (G . (G,DD)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,DD)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,DD)), the carrier of F
FuncZero ( the carrier of (G . (G,DD)),F) is Relation-like the carrier of (G . (G,DD)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,DD)), the carrier of F)
the carrier of (G . (G,DD)) --> (0. F) is non empty Relation-like the carrier of (G . (G,DD)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,DD)), the carrier of F))
K7( the carrier of (G . (G,DD)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,DD)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,DD)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,DD)), the carrier of F)),(Funcs ( the carrier of (G . (G,DD)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,DD)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,DD)), the carrier of F)),(Funcs ( the carrier of (G . (G,DD)), the carrier of F))),(Funcs ( the carrier of (G . (G,DD)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,DD)), the carrier of F)),(Funcs ( the carrier of (G . (G,DD)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,DD)), the carrier of F)),(Funcs ( the carrier of (G . (G,DD)), the carrier of F))),(Funcs ( the carrier of (G . (G,DD)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,DD)), the carrier of F)),(Funcs ( the carrier of (G . (G,DD)), the carrier of F))),(Funcs ( the carrier of (G . (G,DD)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,DD)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,DD)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,DD)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,DD)), the carrier of F))),(Funcs ( the carrier of (G . (G,DD)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,DD)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,DD)), the carrier of F))),(Funcs ( the carrier of (G . (G,DD)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,DD)), the carrier of F))),(Funcs ( the carrier of (G . (G,DD)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,DD)), the carrier of F)),(FuncZero ( the carrier of (G . (G,DD)),F)),(FuncAdd ( the carrier of (G . (G,DD)),F)),(FuncExtMult ( the carrier of (G . (G,DD)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,DD)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,DD)),F)) is set
K215(K266( the carrier of (G . (G,DD)),F),(LinearOperators ((G . (G,DD)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,DD)),F)
K213(K266( the carrier of (G . (G,DD)),F),(LinearOperators ((G . (G,DD)),F))) is Relation-like K7((LinearOperators ((G . (G,DD)),F)),(LinearOperators ((G . (G,DD)),F))) -defined LinearOperators ((G . (G,DD)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,DD)),F)),(LinearOperators ((G . (G,DD)),F))),(LinearOperators ((G . (G,DD)),F))))
K7((LinearOperators ((G . (G,DD)),F)),(LinearOperators ((G . (G,DD)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,DD)),F)),(LinearOperators ((G . (G,DD)),F))),(LinearOperators ((G . (G,DD)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,DD)),F)),(LinearOperators ((G . (G,DD)),F))),(LinearOperators ((G . (G,DD)),F)))) is set
K214(K266( the carrier of (G . (G,DD)),F),(LinearOperators ((G . (G,DD)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,DD)),F))) -defined LinearOperators ((G . (G,DD)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,DD)),F))),(LinearOperators ((G . (G,DD)),F))))
K7(REAL,(LinearOperators ((G . (G,DD)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,DD)),F))),(LinearOperators ((G . (G,DD)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,DD)),F))),(LinearOperators ((G . (G,DD)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,DD)),F)),K215(K266( the carrier of (G . (G,DD)),F),(LinearOperators ((G . (G,DD)),F))),K213(K266( the carrier of (G . (G,DD)),F),(LinearOperators ((G . (G,DD)),F))),K214(K266( the carrier of (G . (G,DD)),F),(LinearOperators ((G . (G,DD)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,DD)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,DD)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))) is Element of BoundedLinearOperators ((G . (G,DD)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))) -defined BoundedLinearOperators ((G . (G,DD)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))),(BoundedLinearOperators ((G . (G,DD)),F))))
K7((BoundedLinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))),(BoundedLinearOperators ((G . (G,DD)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))),(BoundedLinearOperators ((G . (G,DD)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,DD)),F))) -defined BoundedLinearOperators ((G . (G,DD)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,DD)),F))),(BoundedLinearOperators ((G . (G,DD)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,DD)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,DD)),F))),(BoundedLinearOperators ((G . (G,DD)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,DD)),F))),(BoundedLinearOperators ((G . (G,DD)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,DD)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,DD)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,DD)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,DD)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,DD)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,DD)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,DD)),F)),(BoundedLinearOperators ((G . (G,DD)),F))),(BoundedLinearOperatorsNorm ((G . (G,DD)),F)) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)) is non empty set
(G,F,DD,f,X) is Relation-like the carrier of (product G) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)) -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F))))
K7( the carrier of (product G), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)))) is set
(G,F,DD,f,X) /. a is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F))
(G,F,DD,f,X) /. i is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F))
((G,F,DD,f,X) /. a) - ((G,F,DD,f,X) /. i) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F))
- ((G,F,DD,f,X) /. i) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F))
((G,F,DD,f,X) /. a) + (- ((G,F,DD,f,X) /. i)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F))
the addF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)) is Relation-like K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F))) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F))))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F))) is Relation-like set
K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F))) is Relation-like set
K6(K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)))) is set
K462( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)), the addF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)),((G,F,DD,f,X) /. a),(- ((G,F,DD,f,X) /. i))) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F))
||.(((G,F,DD,f,X) /. a) - ((G,F,DD,f,X) /. i)).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)) -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 ((G . (G,DD)),F)),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)),REAL)) is non trivial V49() set
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,DD)),F)) . (((G,F,DD,f,X) /. a) - ((G,F,DD,f,X) /. i)) is V11() real ext-real Element of REAL
y0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
dom y0 is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
min y0 is V11() real ext-real Element of REAL
min_p y0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
y0 . (min_p y0) is V11() real ext-real Element of REAL
r is V11() real ext-real Element of REAL
len y0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
s is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
s - i is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
- i is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
s + (- i) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
the addF of (product G) is Relation-like K7( the carrier of (product G), the carrier of (product G)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)))
K7( the carrier of (product G), the carrier of (product G)) is Relation-like set
K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)) is Relation-like set
K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G))) is set
K462( the carrier of (product G), the addF of (product G),s,(- i)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.(s - i).|| is V11() real ext-real non negative Element of REAL
the normF of (product G) is non empty Relation-like the carrier of (product G) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (product G),REAL))
K7( the carrier of (product G),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (product G),REAL)) is non trivial V49() set
the normF of (product G) . (s - i) is V11() real ext-real Element of REAL
(f `| X) /. s is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
((f `| X) /. s) - ((f `| X) /. i) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
- ((f `| X) /. i) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
((f `| X) /. s) + (- ((f `| X) /. i)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
the addF of (R_NormSpace_of_BoundedLinearOperators ((product G),F)) is Relation-like K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))) is Relation-like set
K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))) is Relation-like set
K6(K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)))) is set
K462( the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)), the addF of (R_NormSpace_of_BoundedLinearOperators ((product G),F)),((f `| X) /. s),(- ((f `| X) /. i))) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
||.(((f `| X) /. s) - ((f `| X) /. i)).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((product G),F)) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)) -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 ((product G),F)),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)),REAL)) is non trivial V49() set
the normF of (R_NormSpace_of_BoundedLinearOperators ((product G),F)) . (((f `| X) /. s) - ((f `| X) /. i)) is V11() real ext-real Element of REAL
diff (f,s) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
diff (f,i) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
(diff (f,s)) - (diff (f,i)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
- (diff (f,i)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
(diff (f,s)) + (- (diff (f,i))) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
K462( the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)), the addF of (R_NormSpace_of_BoundedLinearOperators ((product G),F)),(diff (f,s)),(- (diff (f,i)))) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
y1 is non empty Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like total quasi_total V153( product G,F) V154( product G,F) Lipschitzian Element of K6(K7( the carrier of (product G), the carrier of F))
PreNorms y1 is non empty complex-membered ext-real-membered real-membered Element of K6(REAL)
{ ||.(y1 . b₁).|| where b₁ is Element of the carrier of (product G) : ||.b₁.|| <= 1 } is set
upper_bound (PreNorms y1) is V11() real ext-real Element of REAL
||.((diff (f,s)) - (diff (f,i))).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((product G),F)) . ((diff (f,s)) - (diff (f,i))) is V11() real ext-real Element of REAL
PP is V11() real ext-real set
DD is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
y1 . DD is Element of the carrier of F
||.(y1 . DD).|| is V11() real ext-real non negative Element of REAL
the normF of F is non empty Relation-like the carrier of F -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of F,REAL))
K7( the carrier of F,REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of F,REAL)) is non trivial V49() set
the normF of F . (y1 . DD) is V11() real ext-real Element of REAL
||.DD.|| is V11() real ext-real non negative Element of REAL
the normF of (product G) . DD is V11() real ext-real Element of REAL
(diff (f,i)) . DD is Element of the carrier of F
a is Relation-like NAT -defined the carrier of F -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of F
dom a is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
Sum a is Element of the carrier of F
(diff (f,s)) . DD is Element of the carrier of F
tm is Relation-like NAT -defined the carrier of F -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of F
dom tm is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
Sum tm is Element of the carrier of F
w2 is Relation-like NAT -defined the carrier of F -valued Function-like V49() FinSequence-like FinSubsequence-like countable FinSequence of the carrier of F
len w2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
dom w2 is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
len tm is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
len a is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
ys is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
w2 . ys is set
tm /. ys is Element of the carrier of F
a /. ys is Element of the carrier of F
(tm /. ys) - (a /. ys) is Element of the carrier of F
- (a /. ys) is Element of the carrier of F
(tm /. ys) + (- (a /. ys)) is Element of the carrier of F
the addF of F is Relation-like K7( the carrier of F, the carrier of F) -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of F, the carrier of F), the carrier of F))
K7( the carrier of F, the carrier of F) is Relation-like set
K7(K7( the carrier of F, the carrier of F), the carrier of F) is Relation-like set
K6(K7(K7( the carrier of F, the carrier of F), the carrier of F)) is set
K462( the carrier of F, the addF of F,(tm /. ys),(- (a /. ys))) is Element of the carrier of F
Sum w2 is Element of the carrier of F
(Sum tm) - (Sum a) is Element of the carrier of F
- (Sum a) is Element of the carrier of F
(Sum tm) + (- (Sum a)) is Element of the carrier of F
K462( the carrier of F, the addF of F,(Sum tm),(- (Sum a))) is Element of the carrier of F
||.(Sum w2).|| is V11() real ext-real non negative Element of REAL
the normF of F . (Sum w2) is V11() real ext-real Element of REAL
ys is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
len ys is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
dom ys is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
rm is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
ys . rm is V11() real ext-real Element of REAL
w2 /. rm is Element of the carrier of F
||.(w2 /. rm).|| is V11() real ext-real non negative Element of REAL
the normF of F . (w2 /. rm) 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 complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable FinSequence of REAL
len rs is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
dom rs is V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of K6(NAT)
i is set
rng rs is V49() complex-membered ext-real-membered real-membered countable bounded_below bounded_above real-bounded Element of K6(REAL)
p1 is set
rs . p1 is V11() real ext-real Element of REAL
p0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal countable set
rs . p0 is V11() real ext-real Element of REAL
{rm} is non empty trivial V49() 1 -element complex-membered ext-real-membered real-membered countable left_end right_end bounded_below bounded_above real-bounded Element of K6(REAL)
i is set
rs . 1 is V11() real ext-real Element of REAL
(len G) |-> (i0 / (2 * (len G))) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V49() len G -element FinSequence-like FinSubsequence-like countable Element of (len G) -tuples_on REAL
(len G) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
REAL * is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like V49() FinSequence-like FinSubsequence-like countable Element of REAL * : len b₁ = len G } is set
(Seg (len G)) --> (i0 / (2 * (len G))) is non empty Relation-like NAT -defined Seg (len G) -defined Seg (len G) -defined REAL -valued {(i0 / (2 * (len G)))} -valued Function-like constant total total quasi_total complex-valued ext-real-valued real-valued V49() FinSequence-like FinSubsequence-like countable Element of K6(K7((Seg (len G)),{(i0 / (2 * (len G)))}))
{(i0 / (2 * (len G)))} is non empty trivial V49() 1 -element complex-membered ext-real-membered real-membered countable left_end right_end bounded_below bounded_above real-bounded set
K7((Seg (len G)),{(i0 / (2 * (len G)))}) is Relation-like complex-valued ext-real-valued real-valued V49() countable set
K6(K7((Seg (len G)),{(i0 / (2 * (len G)))})) is V49() V53() countable set
Sum rs is V11() real ext-real Element of REAL
(len G) * (i0 / (2 * (len G))) is V11() real ext-real Element of REAL
i0 / 2 is V11() real ext-real Element of REAL
K39(2) is non empty V11() real ext-real positive non negative set
K37(i0,K39(2)) is V11() real ext-real set
(i0 / 2) / (len G) is V11() real ext-real Element of REAL
K39((len G)) is non empty V11() real ext-real positive non negative set
K37((i0 / 2),K39((len G))) is V11() real ext-real set
(len G) * ((i0 / 2) / (len G)) is V11() real ext-real Element of REAL
i is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered countable bounded_below bounded_above real-bounded Element of NAT
ys . i is V11() real ext-real Element of REAL
w2 /. i is Element of the carrier of F
||.(w2 /. i).|| is V11() real ext-real non negative Element of REAL
the normF of F . (w2 /. i) is V11() real ext-real Element of REAL
w2 . i is set
(G,i) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . (G,i) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (G . (G,i)) is non empty non trivial set
K7( the carrier of (G . (G,i)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,i)), the carrier of F)) is set
(G,F,i,f,s) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,i)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,i)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,i)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,i)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,i)),F))
K266( the carrier of (G . (G,i)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,i)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,i)), the carrier of F
FuncZero ( the carrier of (G . (G,i)),F) is Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,i)), the carrier of F)
the carrier of (G . (G,i)) --> (0. F) is non empty Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,i)), the carrier of F))
FuncAdd ( the carrier of (G . (G,i)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,i)), the carrier of F)),(Funcs ( the carrier of (G . (G,i)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,i)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,i)), the carrier of F)),(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,i)), the carrier of F)),(Funcs ( the carrier of (G . (G,i)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,i)), the carrier of F)),(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,i)), the carrier of F)),(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,i)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,i)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,i)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,i)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,i)), the carrier of F)),(FuncZero ( the carrier of (G . (G,i)),F)),(FuncAdd ( the carrier of (G . (G,i)),F)),(FuncExtMult ( the carrier of (G . (G,i)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,i)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,i)),F)) is set
K215(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,i)),F)
K213(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))) is Relation-like K7((LinearOperators ((G . (G,i)),F)),(LinearOperators ((G . (G,i)),F))) -defined LinearOperators ((G . (G,i)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,i)),F)),(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F))))
K7((LinearOperators ((G . (G,i)),F)),(LinearOperators ((G . (G,i)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,i)),F)),(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,i)),F)),(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F)))) is set
K214(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,i)),F))) -defined LinearOperators ((G . (G,i)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F))))
K7(REAL,(LinearOperators ((G . (G,i)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,i)),F)),K215(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))),K213(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))),K214(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,i)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,i)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))) is Element of BoundedLinearOperators ((G . (G,i)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))) -defined BoundedLinearOperators ((G . (G,i)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F))))
K7((BoundedLinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,i)),F))) -defined BoundedLinearOperators ((G . (G,i)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,i)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,i)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,i)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,i)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,i)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,i)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,i)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperatorsNorm ((G . (G,i)),F)) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)) is non empty set
(G,(G,i),s) is non empty Relation-like the carrier of (G . (G,i)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,i)), the carrier of (product G)))
K7( the carrier of (G . (G,i)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,i)), the carrier of (product G))) is set
f * (G,(G,i),s) is Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,i)), the carrier of F))
(G,(G,i)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,i)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,i))))
K7( the carrier of (product G), the carrier of (G . (G,i))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,i)))) is set
(G,(G,i)) . s is Element of the carrier of (G . (G,i))
diff ((f * (G,(G,i),s)),((G,(G,i)) . s)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
(G,F,i,f,i) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
(G,(G,i),i) is non empty Relation-like the carrier of (G . (G,i)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,i)), the carrier of (product G)))
f * (G,(G,i),i) is Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,i)), the carrier of F))
(G,(G,i)) . i is Element of the carrier of (G . (G,i))
diff ((f * (G,(G,i),i)),((G,(G,i)) . i)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
p1 is non empty Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like total quasi_total V153(G . (G,i),F) V154(G . (G,i),F) Lipschitzian Element of K6(K7( the carrier of (G . (G,i)), the carrier of F))
(G,(G,i)) . DD is Element of the carrier of (G . (G,i))
p1 . ((G,(G,i)) . DD) is Element of the carrier of F
p0 is non empty Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like total quasi_total V153(G . (G,i),F) V154(G . (G,i),F) Lipschitzian Element of K6(K7( the carrier of (G . (G,i)), the carrier of F))
p0 . ((G,(G,i)) . DD) is Element of the carrier of F
a /. i is Element of the carrier of F
a . i is set
tm /. i is Element of the carrier of F
tm . i is set
P0 is Element of the carrier of F
P1 is Element of the carrier of F
P1 - P0 is Element of the carrier of F
- P0 is Element of the carrier of F
P1 + (- P0) is Element of the carrier of F
K462( the carrier of F, the addF of F,P1,(- P0)) is Element of the carrier of F
y0 . i is V11() real ext-real Element of REAL
(G,F,i,f,X) is Relation-like the carrier of (product G) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)) -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))))
K7( the carrier of (product G), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)))) is set
(G,F,i,f,X) /. s is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
(G,F,i,f,X) /. i is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
((G,F,i,f,X) /. s) - ((G,F,i,f,X) /. i) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
- ((G,F,i,f,X) /. i) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
((G,F,i,f,X) /. s) + (- ((G,F,i,f,X) /. i)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
the addF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)) is Relation-like K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))) is Relation-like set
K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))) is Relation-like set
K6(K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)))) is set
K462( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)), the addF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)),((G,F,i,f,X) /. s),(- ((G,F,i,f,X) /. i))) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
||.(((G,F,i,f,X) /. s) - ((G,F,i,f,X) /. i)).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)) -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 ((G . (G,i)),F)),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)),REAL)) is non trivial V49() set
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)) . (((G,F,i,f,X) /. s) - ((G,F,i,f,X) /. i)) is V11() real ext-real Element of REAL
(G,F,i,f,s) - ((G,F,i,f,X) /. i) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
(G,F,i,f,s) + (- ((G,F,i,f,X) /. i)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
K462( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)), the addF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)),(G,F,i,f,s),(- ((G,F,i,f,X) /. i))) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
||.((G,F,i,f,s) - ((G,F,i,f,X) /. i)).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)) . ((G,F,i,f,s) - ((G,F,i,f,X) /. i)) is V11() real ext-real Element of REAL
(G,F,i,f,s) - (G,F,i,f,i) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
- (G,F,i,f,i) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
(G,F,i,f,s) + (- (G,F,i,f,i)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
K462( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)), the addF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)),(G,F,i,f,s),(- (G,F,i,f,i))) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
||.((G,F,i,f,s) - (G,F,i,f,i)).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)) . ((G,F,i,f,s) - (G,F,i,f,i)) is V11() real ext-real Element of REAL
PP is non empty Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like total quasi_total V153(G . (G,i),F) V154(G . (G,i),F) Lipschitzian Element of K6(K7( the carrier of (G . (G,i)), the carrier of F))
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 (G . (G,i)) : ||.b₁.|| <= 1 } is set
upper_bound (PreNorms PP) is V11() real ext-real Element of REAL
pt is Element of the carrier of (G . (G,i))
PP . pt is Element of the carrier of F
||.pt.|| is V11() real ext-real non negative Element of REAL
the normF of (G . (G,i)) is non empty Relation-like the carrier of (G . (G,i)) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (G . (G,i)),REAL))
K7( the carrier of (G . (G,i)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (G . (G,i)),REAL)) is non trivial V49() set
the normF of (G . (G,i)) . pt is V11() real ext-real Element of REAL
||.(PP . pt).|| is V11() real ext-real non negative Element of REAL
the normF of F . (PP . pt) is V11() real ext-real Element of REAL
||.(P1 - P0).|| is V11() real ext-real non negative Element of REAL
the normF of F . (P1 - P0) is V11() real ext-real Element of REAL
rs . i is V11() real ext-real Element of REAL
(diff (f,s)) - ((f `| X) /. i) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
(diff (f,s)) + (- ((f `| X) /. i)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
K462( the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)), the addF of (R_NormSpace_of_BoundedLinearOperators ((product G),F)),(diff (f,s)),(- ((f `| X) /. i))) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
||.((diff (f,s)) - ((f `| X) /. i)).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((product G),F)) . ((diff (f,s)) - ((f `| X) /. i)) is V11() real ext-real Element of REAL
i is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
i is set
(G,i) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V49() cardinal V132() V160() countable Element of dom G
G . (G,i) is non empty non trivial right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
BoundedLinearOperators ((G . (G,i)),F) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,i)),F)))
R_VectorSpace_of_LinearOperators ((G . (G,i)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
LinearOperators ((G . (G,i)),F) is non empty functional Element of K6( the carrier of K266( the carrier of (G . (G,i)),F))
the carrier of (G . (G,i)) is non empty non trivial set
K266( the carrier of (G . (G,i)),F) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital constituted-Functions RLSStruct
Funcs ( the carrier of (G . (G,i)), the carrier of F) is non empty functional FUNCTION_DOMAIN of the carrier of (G . (G,i)), the carrier of F
FuncZero ( the carrier of (G . (G,i)),F) is Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like total quasi_total Element of Funcs ( the carrier of (G . (G,i)), the carrier of F)
the carrier of (G . (G,i)) --> (0. F) is non empty Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like constant total quasi_total Element of K6(K7( the carrier of (G . (G,i)), the carrier of F))
K7( the carrier of (G . (G,i)), the carrier of F) is Relation-like set
K6(K7( the carrier of (G . (G,i)), the carrier of F)) is set
FuncAdd ( the carrier of (G . (G,i)),F) is Relation-like K7((Funcs ( the carrier of (G . (G,i)), the carrier of F)),(Funcs ( the carrier of (G . (G,i)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,i)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((Funcs ( the carrier of (G . (G,i)), the carrier of F)),(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F))))
K7((Funcs ( the carrier of (G . (G,i)), the carrier of F)),(Funcs ( the carrier of (G . (G,i)), the carrier of F))) is Relation-like set
K7(K7((Funcs ( the carrier of (G . (G,i)), the carrier of F)),(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F))) is Relation-like set
K6(K7(K7((Funcs ( the carrier of (G . (G,i)), the carrier of F)),(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F)))) is set
FuncExtMult ( the carrier of (G . (G,i)),F) is Relation-like K7(REAL,(Funcs ( the carrier of (G . (G,i)), the carrier of F))) -defined Funcs ( the carrier of (G . (G,i)), the carrier of F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F))))
K7(REAL,(Funcs ( the carrier of (G . (G,i)), the carrier of F))) is non trivial Relation-like V49() set
K7(K7(REAL,(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(Funcs ( the carrier of (G . (G,i)), the carrier of F))),(Funcs ( the carrier of (G . (G,i)), the carrier of F)))) is non trivial V49() set
RLSStruct(# (Funcs ( the carrier of (G . (G,i)), the carrier of F)),(FuncZero ( the carrier of (G . (G,i)),F)),(FuncAdd ( the carrier of (G . (G,i)),F)),(FuncExtMult ( the carrier of (G . (G,i)),F)) #) is strict RLSStruct
the carrier of K266( the carrier of (G . (G,i)),F) is non empty set
K6( the carrier of K266( the carrier of (G . (G,i)),F)) is set
K215(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))) is Relation-like Function-like Element of LinearOperators ((G . (G,i)),F)
K213(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))) is Relation-like K7((LinearOperators ((G . (G,i)),F)),(LinearOperators ((G . (G,i)),F))) -defined LinearOperators ((G . (G,i)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7((LinearOperators ((G . (G,i)),F)),(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F))))
K7((LinearOperators ((G . (G,i)),F)),(LinearOperators ((G . (G,i)),F))) is Relation-like set
K7(K7((LinearOperators ((G . (G,i)),F)),(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F))) is Relation-like set
K6(K7(K7((LinearOperators ((G . (G,i)),F)),(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F)))) is set
K214(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))) is Relation-like K7(REAL,(LinearOperators ((G . (G,i)),F))) -defined LinearOperators ((G . (G,i)),F) -valued Function-like total quasi_total Function-yielding V134() Element of K6(K7(K7(REAL,(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F))))
K7(REAL,(LinearOperators ((G . (G,i)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(LinearOperators ((G . (G,i)),F))),(LinearOperators ((G . (G,i)),F)))) is non trivial V49() set
RLSStruct(# (LinearOperators ((G . (G,i)),F)),K215(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))),K213(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))),K214(K266( the carrier of (G . (G,i)),F),(LinearOperators ((G . (G,i)),F))) #) is right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,i)),F)) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((G . (G,i)),F))) is set
K215((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))) is Element of BoundedLinearOperators ((G . (G,i)),F)
K213((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))) is Relation-like K7((BoundedLinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))) -defined BoundedLinearOperators ((G . (G,i)),F) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F))))
K7((BoundedLinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))) is Relation-like set
K7(K7((BoundedLinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F))) is Relation-like set
K6(K7(K7((BoundedLinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F)))) is set
K214((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))) is Relation-like K7(REAL,(BoundedLinearOperators ((G . (G,i)),F))) -defined BoundedLinearOperators ((G . (G,i)),F) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F))))
K7(REAL,(BoundedLinearOperators ((G . (G,i)),F))) is non trivial Relation-like V49() set
K7(K7(REAL,(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F))) is non trivial Relation-like V49() set
K6(K7(K7(REAL,(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperators ((G . (G,i)),F)))) is non trivial V49() set
BoundedLinearOperatorsNorm ((G . (G,i)),F) is non empty Relation-like BoundedLinearOperators ((G . (G,i)),F) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((G . (G,i)),F)),REAL))
K7((BoundedLinearOperators ((G . (G,i)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7((BoundedLinearOperators ((G . (G,i)),F)),REAL)) is non trivial V49() set
NORMSTR(# (BoundedLinearOperators ((G . (G,i)),F)),K215((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),K213((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),K214((R_VectorSpace_of_LinearOperators ((G . (G,i)),F)),(BoundedLinearOperators ((G . (G,i)),F))),(BoundedLinearOperatorsNorm ((G . (G,i)),F)) #) is strict NORMSTR
(G,F,i,f,X) is Relation-like the carrier of (product G) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)) -valued Function-like Element of K6(K7( the carrier of (product G), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))))
the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)) is non empty set
K7( the carrier of (product G), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)))) is set
y0 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,F,i,f,y0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
(G,(G,i),y0) is non empty Relation-like the carrier of (G . (G,i)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,i)), the carrier of (product G)))
K7( the carrier of (G . (G,i)), the carrier of (product G)) is Relation-like set
K6(K7( the carrier of (G . (G,i)), the carrier of (product G))) is set
f * (G,(G,i),y0) is Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,i)), the carrier of F))
(G,(G,i)) is non empty Relation-like the carrier of (product G) -defined the carrier of (G . (G,i)) -valued Function-like total quasi_total Element of K6(K7( the carrier of (product G), the carrier of (G . (G,i))))
K7( the carrier of (product G), the carrier of (G . (G,i))) is Relation-like set
K6(K7( the carrier of (product G), the carrier of (G . (G,i)))) is set
(G,(G,i)) . y0 is Element of the carrier of (G . (G,i))
diff ((f * (G,(G,i),y0)),((G,(G,i)) . y0)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
0. (product G) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like V68( product G) countable Element of the carrier of (product G)
the ZeroF of (product G) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,(G,i),(0. (product G))) is non empty Relation-like the carrier of (G . (G,i)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,i)), the carrier of (product G)))
diff (f,y0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
(G,(G,i),(0. (product G))) (#) (diff (f,y0)) is Relation-like the carrier of (G . (G,i)) -defined Function-like set
y0 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
dom (G,F,i,f,X) is Element of K6( the carrier of (product G))
y0 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
(G,F,i,f,X) /. y0 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
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 ((product G),F))
s is V11() real ext-real Element of REAL
y1 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
y1 - y0 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
- y0 is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
y1 + (- y0) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
the addF of (product G) is Relation-like K7( the carrier of (product G), the carrier of (product G)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)))
K7( the carrier of (product G), the carrier of (product G)) is Relation-like set
K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G)) is Relation-like set
K6(K7(K7( the carrier of (product G), the carrier of (product G)), the carrier of (product G))) is set
K462( the carrier of (product G), the addF of (product G),y1,(- y0)) is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.(y1 - y0).|| is V11() real ext-real non negative Element of REAL
the normF of (product G) is non empty Relation-like the carrier of (product G) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (product G),REAL))
K7( the carrier of (product G),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (product G),REAL)) is non trivial V49() set
the normF of (product G) . (y1 - y0) is V11() real ext-real Element of REAL
(G,F,i,f,X) /. y1 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
((G,F,i,f,X) /. y1) - ((G,F,i,f,X) /. y0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
- ((G,F,i,f,X) /. y0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
((G,F,i,f,X) /. y1) + (- ((G,F,i,f,X) /. y0)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
the addF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)) is Relation-like K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))) is Relation-like set
K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))) is Relation-like set
K6(K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)))) is set
K462( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)), the addF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)),((G,F,i,f,X) /. y1),(- ((G,F,i,f,X) /. y0))) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
||.(((G,F,i,f,X) /. y1) - ((G,F,i,f,X) /. y0)).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)) -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 ((G . (G,i)),F)),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)),REAL)) is non trivial V49() set
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)) . (((G,F,i,f,X) /. y1) - ((G,F,i,f,X) /. 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 ((product G),F))
((f `| X) /. y1) - ((f `| X) /. y0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
- ((f `| X) /. y0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
((f `| X) /. y1) + (- ((f `| X) /. y0)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
the addF of (R_NormSpace_of_BoundedLinearOperators ((product G),F)) is Relation-like K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)) -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))) is Relation-like set
K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))) is Relation-like set
K6(K7(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)), the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))), the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)))) is set
K462( the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)), the addF of (R_NormSpace_of_BoundedLinearOperators ((product G),F)),((f `| X) /. y1),(- ((f `| X) /. y0))) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((product G),F)) is non empty Relation-like the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)) -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 ((product G),F)),REAL))
K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)),REAL)) is non trivial V49() set
the normF of (R_NormSpace_of_BoundedLinearOperators ((product G),F)) . (((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 ((product G),F))
(diff (f,y1)) - ((f `| X) /. y0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
(diff (f,y1)) + (- ((f `| X) /. y0)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
K462( the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)), the addF of (R_NormSpace_of_BoundedLinearOperators ((product G),F)),(diff (f,y1)),(- ((f `| X) /. y0))) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
||.((diff (f,y1)) - ((f `| X) /. y0)).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((product G),F)) . ((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 ((product G),F))
(diff (f,y1)) - (diff (f,y0)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
- (diff (f,y0)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
(diff (f,y1)) + (- (diff (f,y0))) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
K462( the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F)), the addF of (R_NormSpace_of_BoundedLinearOperators ((product G),F)),(diff (f,y1)),(- (diff (f,y0)))) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((product G),F))
||.((diff (f,y1)) - (diff (f,y0))).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((product G),F)) . ((diff (f,y1)) - (diff (f,y0))) is V11() real ext-real Element of REAL
(G,F,i,f,y1) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
(G,(G,i),y1) is non empty Relation-like the carrier of (G . (G,i)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,i)), the carrier of (product G)))
f * (G,(G,i),y1) is Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,i)), the carrier of F))
(G,(G,i)) . y1 is Element of the carrier of (G . (G,i))
diff ((f * (G,(G,i),y1)),((G,(G,i)) . y1)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
(G,(G,i),(0. (product G))) (#) (diff (f,y1)) is Relation-like the carrier of (G . (G,i)) -defined Function-like set
(G,F,i,f,y0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
(G,(G,i),y0) is non empty Relation-like the carrier of (G . (G,i)) -defined the carrier of (product G) -valued Function-like total quasi_total Element of K6(K7( the carrier of (G . (G,i)), the carrier of (product G)))
f * (G,(G,i),y0) is Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like Element of K6(K7( the carrier of (G . (G,i)), the carrier of F))
(G,(G,i)) . y0 is Element of the carrier of (G . (G,i))
diff ((f * (G,(G,i),y0)),((G,(G,i)) . y0)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
(G,(G,i),(0. (product G))) (#) (diff (f,y0)) is Relation-like the carrier of (G . (G,i)) -defined Function-like set
(G,F,i,f,y1) - (G,F,i,f,y0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
- (G,F,i,f,y0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
(G,F,i,f,y1) + (- (G,F,i,f,y0)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
K462( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)), the addF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)),(G,F,i,f,y1),(- (G,F,i,f,y0))) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
PP is non empty Relation-like the carrier of (G . (G,i)) -defined the carrier of F -valued Function-like total quasi_total V153(G . (G,i),F) V154(G . (G,i),F) Lipschitzian Element of K6(K7( the carrier of (G . (G,i)), the carrier of F))
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 (G . (G,i)) : ||.b₁.|| <= 1 } is set
upper_bound (PreNorms PP) is V11() real ext-real Element of REAL
||.((G,F,i,f,y1) - (G,F,i,f,y0)).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)) . ((G,F,i,f,y1) - (G,F,i,f,y0)) is V11() real ext-real Element of REAL
DD is non empty Relation-like the carrier of (product G) -defined the carrier of F -valued Function-like total quasi_total V153( product G,F) V154( product G,F) Lipschitzian Element of K6(K7( the carrier of (product G), the carrier of F))
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 (product G) : ||.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 (G . (G,i))
PP . t is Element of the carrier of F
||.(PP . t).|| is V11() real ext-real non negative Element of REAL
the normF of F is non empty Relation-like the carrier of F -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of F,REAL))
K7( the carrier of F,REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of F,REAL)) is non trivial V49() set
the normF of F . (PP . t) is V11() real ext-real Element of REAL
||.t.|| is V11() real ext-real non negative Element of REAL
the normF of (G . (G,i)) is non empty Relation-like the carrier of (G . (G,i)) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (G . (G,i)),REAL))
K7( the carrier of (G . (G,i)),REAL) is non trivial Relation-like complex-valued ext-real-valued real-valued V49() set
K6(K7( the carrier of (G . (G,i)),REAL)) is non trivial V49() set
the normF of (G . (G,i)) . t is V11() real ext-real Element of REAL
dom (G,(G,i),(0. (product G))) is non empty Element of K6( the carrier of (G . (G,i)))
K6( the carrier of (G . (G,i))) is set
(G,(G,i),(0. (product G))) . t is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
tm is Relation-like NAT -defined Function-like V49() len G -element FinSequence-like FinSubsequence-like countable Element of the carrier of (product G)
||.tm.|| is V11() real ext-real non negative Element of REAL
the normF of (product G) . tm is V11() real ext-real Element of REAL
(G,F,i,f,y1) . t is Element of the carrier of F
(diff (f,y1)) . tm is Element of the carrier of F
(G,F,i,f,y0) . t is Element of the carrier of F
(diff (f,y0)) . tm is Element of the carrier of F
((diff (f,y1)) . tm) - ((diff (f,y0)) . tm) is Element of the carrier of F
- ((diff (f,y0)) . tm) is Element of the carrier of F
((diff (f,y1)) . tm) + (- ((diff (f,y0)) . tm)) is Element of the carrier of F
the addF of F is Relation-like K7( the carrier of F, the carrier of F) -defined the carrier of F -valued Function-like total quasi_total Element of K6(K7(K7( the carrier of F, the carrier of F), the carrier of F))
K7( the carrier of F, the carrier of F) is Relation-like set
K7(K7( the carrier of F, the carrier of F), the carrier of F) is Relation-like set
K6(K7(K7( the carrier of F, the carrier of F), the carrier of F)) is set
K462( the carrier of F, the addF of F,((diff (f,y1)) . tm),(- ((diff (f,y0)) . tm))) is Element of the carrier of F
||.(((diff (f,y1)) . tm) - ((diff (f,y0)) . tm)).|| is V11() real ext-real non negative Element of REAL
the normF of F . (((diff (f,y1)) . tm) - ((diff (f,y0)) . tm)) is V11() real ext-real Element of REAL
DD . tm is Element of the carrier of F
||.(DD . tm).|| is V11() real ext-real non negative Element of REAL
the normF of F . (DD . tm) is V11() real ext-real Element of REAL
(G,F,i,f,y1) - ((G,F,i,f,X) /. y0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
(G,F,i,f,y1) + (- ((G,F,i,f,X) /. y0)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
K462( the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)), the addF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)),(G,F,i,f,y1),(- ((G,F,i,f,X) /. y0))) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F))
||.((G,F,i,f,y1) - ((G,F,i,f,X) /. y0)).|| is V11() real ext-real non negative Element of REAL
the normF of (R_NormSpace_of_BoundedLinearOperators ((G . (G,i)),F)) . ((G,F,i,f,y1) - ((G,F,i,f,X) /. y0)) is V11() real ext-real Element of REAL