:: NDIFF_1 semantic presentation

REAL is V1() V49() V170() V171() V172() V176() set
NAT is V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() Element of K6(REAL)
K6(REAL) is set
COMPLEX is V1() V49() V170() V176() set
K7(REAL,REAL) is complex-valued ext-real-valued real-valued set
K6(K7(REAL,REAL)) is set
omega is V1() epsilon-transitive epsilon-connected ordinal V170() V171() V172() V173() V174() V175() V176() set
K6(omega) is set
K6(NAT) is set
K7(NAT,REAL) is complex-valued ext-real-valued real-valued set
K6(K7(NAT,REAL)) is set
the_set_of_RealSequences is V1() set
K7(the_set_of_RealSequences,the_set_of_RealSequences) is set
K7(K7(the_set_of_RealSequences,the_set_of_RealSequences),the_set_of_RealSequences) is set
K6(K7(K7(the_set_of_RealSequences,the_set_of_RealSequences),the_set_of_RealSequences)) is set
K7(REAL,the_set_of_RealSequences) is set
K7(K7(REAL,the_set_of_RealSequences),the_set_of_RealSequences) is set
K6(K7(K7(REAL,the_set_of_RealSequences),the_set_of_RealSequences)) is set
Linear_Space_of_RealSequences is RLSStruct
the carrier of Linear_Space_of_RealSequences is set
K6( the carrier of Linear_Space_of_RealSequences) is set
the_set_of_l2RealSequences is Element of K6( the carrier of Linear_Space_of_RealSequences)
K7(the_set_of_l2RealSequences,the_set_of_l2RealSequences) is set
K7(K7(the_set_of_l2RealSequences,the_set_of_l2RealSequences),REAL) is complex-valued ext-real-valued real-valued set
K6(K7(K7(the_set_of_l2RealSequences,the_set_of_l2RealSequences),REAL)) is set
RAT is V1() V49() V170() V171() V172() V173() V176() set
INT is V1() V49() V170() V171() V172() V173() V174() V176() set
K7(COMPLEX,COMPLEX) is complex-valued set
K6(K7(COMPLEX,COMPLEX)) is set
K7(K7(COMPLEX,COMPLEX),COMPLEX) is complex-valued set
K6(K7(K7(COMPLEX,COMPLEX),COMPLEX)) is set
K7(K7(REAL,REAL),REAL) is complex-valued ext-real-valued real-valued set
K6(K7(K7(REAL,REAL),REAL)) is set
K7(RAT,RAT) is V20( RAT ) complex-valued ext-real-valued real-valued set
K6(K7(RAT,RAT)) is set
K7(K7(RAT,RAT),RAT) is V20( RAT ) complex-valued ext-real-valued real-valued set
K6(K7(K7(RAT,RAT),RAT)) is set
K7(INT,INT) is V20( RAT ) V20( INT ) complex-valued ext-real-valued real-valued set
K6(K7(INT,INT)) is set
K7(K7(INT,INT),INT) is V20( RAT ) V20( INT ) complex-valued ext-real-valued real-valued set
K6(K7(K7(INT,INT),INT)) is set
K7(NAT,NAT) is V20( RAT ) V20( INT ) complex-valued ext-real-valued real-valued natural-valued set
K7(K7(NAT,NAT),NAT) is V20( RAT ) V20( INT ) complex-valued ext-real-valued real-valued natural-valued set
K6(K7(K7(NAT,NAT),NAT)) is set
0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
the V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V170() V171() V172() V173() V174() V175() V176() set is V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V170() V171() V172() V173() V174() V175() V176() set
0c is set
1 is V1() epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
- 1 is V11() real ext-real non positive Element of REAL
K39(0) is V11() real ext-real set
T is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() set
S is Element of the carrier of T
f is Neighbourhood of S
x0 is Neighbourhood of S
N is V11() real ext-real Element of REAL
{ b₁ where b₁ is Element of the carrier of T : not N <= ||.(b₁ - S).|| } is set
L is V11() real ext-real Element of REAL
{ b₁ where b₁ is Element of the carrier of T : not L <= ||.(b₁ - S).|| } is set
min (N,L) is V11() real ext-real Element of REAL
{ b₁ where b₁ is Element of the carrier of T : not min (N,L) <= ||.(b₁ - S).|| } is set
K6( the carrier of T) is set
s1 is set
N2 is Element of the carrier of T
N2 - S is Element of the carrier of T
- S is Element of the carrier of T
K192(T,N2,(- S)) is Element of the carrier of T
||.(N2 - S).|| is V11() real ext-real Element of REAL
s1 is set
N2 is Element of the carrier of T
N2 - S is Element of the carrier of T
- S is Element of the carrier of T
K192(T,N2,(- S)) is Element of the carrier of T
||.(N2 - S).|| is V11() real ext-real Element of REAL
T is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() set
K6( the carrier of T) is set
S is Element of K6( the carrier of T)
S ` is Element of K6( the carrier of T)
K4( the carrier of T,S) is set
f is Element of the carrier of T
{ b₁ where b₁ is Element of the carrier of T : not 1 / (a₁ + 1) <= ||.(b₁ - f).|| } is set
x0 is V11() real ext-real Element of REAL
{ b₁ where b₁ is Element of the carrier of T : not x0 <= ||.(b₁ - f).|| } is set
L is set
R is Element of the carrier of T
R - f is Element of the carrier of T
- f is Element of the carrier of T
K192(T,R,(- f)) is Element of the carrier of T
||.(R - f).|| is V11() real ext-real Element of REAL
s1 is Element of the carrier of T
x0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
x0 + 1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
1 / (x0 + 1) is V11() real ext-real Element of REAL
{ b₁ where b₁ is Element of the carrier of T : not 1 / (x0 + 1) <= ||.(b₁ - f).|| } is set
(x0 + 1) " is V11() real ext-real Element of REAL
1 * ((x0 + 1) ") is V11() real ext-real Element of REAL
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
x0 is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
rng x0 is Element of K6( the carrier of T)
N is set
dom x0 is V170() V171() V172() V173() V174() V175() Element of K6(NAT)
L is set
x0 . L is set
R is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
x0 . R is Element of the carrier of T
N is V11() real ext-real Element of REAL
N " is V11() real ext-real Element of REAL
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
L + 1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
(N ") + 0 is V11() real ext-real Element of REAL
1 / (N ") is V11() real ext-real Element of REAL
1 / (L + 1) is V11() real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
s1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
R + 1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
s1 + 1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
x0 . s1 is Element of the carrier of T
1 / (s1 + 1) is V11() real ext-real Element of REAL
{ b₁ where b₁ is Element of the carrier of T : not 1 / (s1 + 1) <= ||.(b₁ - f).|| } is set
1 / (R + 1) is V11() real ext-real Element of REAL
(x0 . s1) - f is Element of the carrier of T
K192(T,(x0 . s1),(- f)) is Element of the carrier of T
||.((x0 . s1) - f).|| is V11() real ext-real Element of REAL
N2 is Element of the carrier of T
N2 - f is Element of the carrier of T
K192(T,N2,(- f)) is Element of the carrier of T
||.(N2 - f).|| is V11() real ext-real Element of REAL
lim x0 is Element of the carrier of T
T is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() set
K6( the carrier of T) is set
S is Element of K6( the carrier of T)
f is Element of the carrier of T
x0 is Neighbourhood of f
N is V11() real ext-real Element of REAL
{ b₁ where b₁ is Element of the carrier of T : not N <= ||.(b₁ - f).|| } is set
T is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() set
K6( the carrier of T) is set
S is Element of K6( the carrier of T)
S ` is Element of K6( the carrier of T)
K4( the carrier of T,S) is set
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
f is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
rng f is Element of K6( the carrier of T)
lim f is Element of the carrier of T
x0 is Neighbourhood of lim f
N is V11() real ext-real Element of REAL
{ b₁ where b₁ is Element of the carrier of T : not N <= ||.(b₁ - (lim f)).|| } is set
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
dom f is V170() V171() V172() V173() V174() V175() Element of K6(NAT)
f . L is Element of the carrier of T
(f . L) - (lim f) is Element of the carrier of T
- (lim f) is Element of the carrier of T
K192(T,(f . L),(- (lim f))) is Element of the carrier of T
||.((f . L) - (lim f)).|| is V11() real ext-real Element of REAL
T is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() set
K6( the carrier of T) is set
f is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of f is V1() set
K6( the carrier of f) is set
S is Element of K6( the carrier of T)
x0 is Element of K6( the carrier of f)
N is Element of the carrier of f
T is set
S is ZeroStr
the carrier of S is set
K7(T, the carrier of S) is set
K6(K7(T, the carrier of S)) is set
f is V16() V19(T) V20( the carrier of S) Function-like quasi_total Element of K6(K7(T, the carrier of S))
rng f is Element of K6( the carrier of S)
K6( the carrier of S) is set
NonZero S is Element of K6( the carrier of S)
[#] S is non proper Element of K6( the carrier of S)
0. S is V68(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
{(0. S)} is set
([#] S) \ {(0. S)} is Element of K6( the carrier of S)
T is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() set
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
S is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
rng S is Element of K6( the carrier of T)
K6( the carrier of T) is set
NonZero T is Element of K6( the carrier of T)
[#] T is V1() non proper Element of K6( the carrier of T)
{(0. T)} is set
([#] T) \ {(0. T)} is Element of K6( the carrier of T)
f is set
S . f is set
dom S is V170() V171() V172() V173() V174() V175() Element of K6(NAT)
f is set
rng S is Element of K6( the carrier of T)
K6( the carrier of T) is set
dom S is V170() V171() V172() V173() V174() V175() Element of K6(NAT)
x0 is set
S . x0 is set
{(0. T)} is set
NonZero T is Element of K6( the carrier of T)
[#] T is V1() non proper Element of K6( the carrier of T)
([#] T) \ {(0. T)} is Element of K6( the carrier of T)
T is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() set
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
S is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
S . f is Element of the carrier of T
f is set
S . f is set
T is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of T is V1() set
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
S is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
f is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
x0 is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
N is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
x0 . L is Element of the carrier of T
N . L is Element of the carrier of T
S . L is Element of the carrier of T
f . L is V11() real ext-real Element of REAL
(f . L) * (S . L) is Element of the carrier of T
the Mult of T is V16() V19(K7(REAL, the carrier of T)) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of T), the carrier of T))
K7(REAL, the carrier of T) is set
K7(K7(REAL, the carrier of T), the carrier of T) is set
K6(K7(K7(REAL, the carrier of T), the carrier of T)) is set
K249( the Mult of T,(f . L),(S . L)) is set
T is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of T is V1() set
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
S is Element of the carrier of T
f is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
x0 is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
N is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
x0 . L is Element of the carrier of T
N . L is Element of the carrier of T
f . L is V11() real ext-real Element of REAL
(f . L) * S is Element of the carrier of T
the Mult of T is V16() V19(K7(REAL, the carrier of T)) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of T), the carrier of T))
K7(REAL, the carrier of T) is set
K7(K7(REAL, the carrier of T), the carrier of T) is set
K6(K7(K7(REAL, the carrier of T), the carrier of T)) is set
K249( the Mult of T,(f . L),S) is set
T is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() set
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
S is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
f is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
x0 is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
f + x0 is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(T,S,(f + x0)) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(T,S,f) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(T,S,x0) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(T,S,f) + (T,S,x0) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
N is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
(T,S,(f + x0)) . N is Element of the carrier of T
S . N is Element of the carrier of T
(f + x0) . N is V11() real ext-real Element of REAL
((f + x0) . N) * (S . N) is Element of the carrier of T
the Mult of T is V16() V19(K7(REAL, the carrier of T)) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of T), the carrier of T))
K7(REAL, the carrier of T) is set
K7(K7(REAL, the carrier of T), the carrier of T) is set
K6(K7(K7(REAL, the carrier of T), the carrier of T)) is set
K249( the Mult of T,((f + x0) . N),(S . N)) is set
f . N is V11() real ext-real Element of REAL
x0 . N is V11() real ext-real Element of REAL
(f . N) + (x0 . N) is V11() real ext-real Element of REAL
((f . N) + (x0 . N)) * (S . N) is Element of the carrier of T
K249( the Mult of T,((f . N) + (x0 . N)),(S . N)) is set
(f . N) * (S . N) is Element of the carrier of T
K249( the Mult of T,(f . N),(S . N)) is set
(x0 . N) * (S . N) is Element of the carrier of T
K249( the Mult of T,(x0 . N),(S . N)) is set
((f . N) * (S . N)) + ((x0 . N) * (S . N)) is Element of the carrier of T
(T,S,f) . N is Element of the carrier of T
((T,S,f) . N) + ((x0 . N) * (S . N)) is Element of the carrier of T
(T,S,x0) . N is Element of the carrier of T
((T,S,f) . N) + ((T,S,x0) . N) is Element of the carrier of T
((T,S,f) + (T,S,x0)) . N is Element of the carrier of T
T is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() set
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
S is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
f is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
x0 is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
f + x0 is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(T,(f + x0),S) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(T,f,S) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(T,x0,S) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(T,f,S) + (T,x0,S) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
N is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
(T,(f + x0),S) . N is Element of the carrier of T
(f + x0) . N is Element of the carrier of T
S . N is V11() real ext-real Element of REAL
(S . N) * ((f + x0) . N) is Element of the carrier of T
the Mult of T is V16() V19(K7(REAL, the carrier of T)) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of T), the carrier of T))
K7(REAL, the carrier of T) is set
K7(K7(REAL, the carrier of T), the carrier of T) is set
K6(K7(K7(REAL, the carrier of T), the carrier of T)) is set
K249( the Mult of T,(S . N),((f + x0) . N)) is set
f . N is Element of the carrier of T
x0 . N is Element of the carrier of T
(f . N) + (x0 . N) is Element of the carrier of T
(S . N) * ((f . N) + (x0 . N)) is Element of the carrier of T
K249( the Mult of T,(S . N),((f . N) + (x0 . N))) is set
(S . N) * (f . N) is Element of the carrier of T
K249( the Mult of T,(S . N),(f . N)) is set
(S . N) * (x0 . N) is Element of the carrier of T
K249( the Mult of T,(S . N),(x0 . N)) is set
((S . N) * (f . N)) + ((S . N) * (x0 . N)) is Element of the carrier of T
(T,f,S) . N is Element of the carrier of T
((T,f,S) . N) + ((S . N) * (x0 . N)) is Element of the carrier of T
(T,x0,S) . N is Element of the carrier of T
((T,f,S) . N) + ((T,x0,S) . N) is Element of the carrier of T
((T,f,S) + (T,x0,S)) . N is Element of the carrier of T
T is V11() real ext-real Element of REAL
S is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() set
K7(NAT, the carrier of S) is set
K6(K7(NAT, the carrier of S)) is set
f is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
T * f is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
x0 is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(S,f,x0) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
T * (S,f,x0) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
(S,(T * f),x0) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
N is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
(T * (S,f,x0)) . N is Element of the carrier of S
(S,f,x0) . N is Element of the carrier of S
T * ((S,f,x0) . N) is Element of the carrier of S
the Mult of S is V16() V19(K7(REAL, the carrier of S)) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of S), the carrier of S))
K7(REAL, the carrier of S) is set
K7(K7(REAL, the carrier of S), the carrier of S) is set
K6(K7(K7(REAL, the carrier of S), the carrier of S)) is set
K249( the Mult of S,T,((S,f,x0) . N)) is set
f . N is Element of the carrier of S
x0 . N is V11() real ext-real Element of REAL
(x0 . N) * (f . N) is Element of the carrier of S
K249( the Mult of S,(x0 . N),(f . N)) is set
T * ((x0 . N) * (f . N)) is Element of the carrier of S
K249( the Mult of S,T,((x0 . N) * (f . N))) is set
T * (x0 . N) is V11() real ext-real Element of REAL
(T * (x0 . N)) * (f . N) is Element of the carrier of S
K249( the Mult of S,(T * (x0 . N)),(f . N)) is set
T * (f . N) is Element of the carrier of S
K249( the Mult of S,T,(f . N)) is set
(x0 . N) * (T * (f . N)) is Element of the carrier of S
K249( the Mult of S,(x0 . N),(T * (f . N))) is set
(T * f) . N is Element of the carrier of S
(x0 . N) * ((T * f) . N) is Element of the carrier of S
K249( the Mult of S,(x0 . N),((T * f) . N)) is set
(S,(T * f),x0) . N is Element of the carrier of S
T is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() set
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
S is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
f is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
x0 is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
f - x0 is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
- x0 is V16() V19( NAT ) Function-like total complex-valued ext-real-valued real-valued set
K38(1) is V11() real ext-real non positive set
K38(1) (#) x0 is V16() V19( NAT ) Function-like total complex-valued ext-real-valued real-valued set
f + (- x0) is V16() V19( NAT ) Function-like total complex-valued ext-real-valued real-valued set
(T,S,(f - x0)) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(T,S,f) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(T,S,x0) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(T,S,f) - (T,S,x0) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
N is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
(T,S,(f - x0)) . N is Element of the carrier of T
S . N is Element of the carrier of T
- x0 is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
f + (- x0) is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(f + (- x0)) . N is V11() real ext-real Element of REAL
((f + (- x0)) . N) * (S . N) is Element of the carrier of T
the Mult of T is V16() V19(K7(REAL, the carrier of T)) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of T), the carrier of T))
K7(REAL, the carrier of T) is set
K7(K7(REAL, the carrier of T), the carrier of T) is set
K6(K7(K7(REAL, the carrier of T), the carrier of T)) is set
K249( the Mult of T,((f + (- x0)) . N),(S . N)) is set
f . N is V11() real ext-real Element of REAL
(- x0) . N is V11() real ext-real Element of REAL
(f . N) + ((- x0) . N) is V11() real ext-real Element of REAL
((f . N) + ((- x0) . N)) * (S . N) is Element of the carrier of T
K249( the Mult of T,((f . N) + ((- x0) . N)),(S . N)) is set
x0 . N is V11() real ext-real Element of REAL
- (x0 . N) is V11() real ext-real Element of REAL
(f . N) + (- (x0 . N)) is V11() real ext-real Element of REAL
((f . N) + (- (x0 . N))) * (S . N) is Element of the carrier of T
K249( the Mult of T,((f . N) + (- (x0 . N))),(S . N)) is set
(f . N) - (x0 . N) is V11() real ext-real Element of REAL
((f . N) - (x0 . N)) * (S . N) is Element of the carrier of T
K249( the Mult of T,((f . N) - (x0 . N)),(S . N)) is set
(f . N) * (S . N) is Element of the carrier of T
K249( the Mult of T,(f . N),(S . N)) is set
(x0 . N) * (S . N) is Element of the carrier of T
K249( the Mult of T,(x0 . N),(S . N)) is set
((f . N) * (S . N)) - ((x0 . N) * (S . N)) is Element of the carrier of T
- ((x0 . N) * (S . N)) is Element of the carrier of T
K192(T,((f . N) * (S . N)),(- ((x0 . N) * (S . N)))) is Element of the carrier of T
(T,S,f) . N is Element of the carrier of T
((T,S,f) . N) - ((x0 . N) * (S . N)) is Element of the carrier of T
K192(T,((T,S,f) . N),(- ((x0 . N) * (S . N)))) is Element of the carrier of T
(T,S,x0) . N is Element of the carrier of T
((T,S,f) . N) - ((T,S,x0) . N) is Element of the carrier of T
- ((T,S,x0) . N) is Element of the carrier of T
K192(T,((T,S,f) . N),(- ((T,S,x0) . N))) is Element of the carrier of T
((T,S,f) - (T,S,x0)) . N is Element of the carrier of T
T is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() set
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
S is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
f is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
x0 is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
f - x0 is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(T,(f - x0),S) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(T,f,S) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(T,x0,S) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(T,f,S) - (T,x0,S) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
N is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
(T,(f - x0),S) . N is Element of the carrier of T
(f - x0) . N is Element of the carrier of T
S . N is V11() real ext-real Element of REAL
(S . N) * ((f - x0) . N) is Element of the carrier of T
the Mult of T is V16() V19(K7(REAL, the carrier of T)) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of T), the carrier of T))
K7(REAL, the carrier of T) is set
K7(K7(REAL, the carrier of T), the carrier of T) is set
K6(K7(K7(REAL, the carrier of T), the carrier of T)) is set
K249( the Mult of T,(S . N),((f - x0) . N)) is set
f . N is Element of the carrier of T
x0 . N is Element of the carrier of T
(f . N) - (x0 . N) is Element of the carrier of T
- (x0 . N) is Element of the carrier of T
K192(T,(f . N),(- (x0 . N))) is Element of the carrier of T
(S . N) * ((f . N) - (x0 . N)) is Element of the carrier of T
K249( the Mult of T,(S . N),((f . N) - (x0 . N))) is set
(S . N) * (f . N) is Element of the carrier of T
K249( the Mult of T,(S . N),(f . N)) is set
(S . N) * (x0 . N) is Element of the carrier of T
K249( the Mult of T,(S . N),(x0 . N)) is set
((S . N) * (f . N)) - ((S . N) * (x0 . N)) is Element of the carrier of T
- ((S . N) * (x0 . N)) is Element of the carrier of T
K192(T,((S . N) * (f . N)),(- ((S . N) * (x0 . N)))) is Element of the carrier of T
(T,f,S) . N is Element of the carrier of T
((T,f,S) . N) - ((S . N) * (x0 . N)) is Element of the carrier of T
K192(T,((T,f,S) . N),(- ((S . N) * (x0 . N)))) is Element of the carrier of T
(T,x0,S) . N is Element of the carrier of T
((T,f,S) . N) - ((T,x0,S) . N) is Element of the carrier of T
- ((T,x0,S) . N) is Element of the carrier of T
K192(T,((T,f,S) . N),(- ((T,x0,S) . N))) is Element of the carrier of T
((T,f,S) - (T,x0,S)) . N is Element of the carrier of T
T is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() set
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
S is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
f is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(T,f,S) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
x0 is V11() real ext-real set
N is Element of the carrier of T
L is V11() real ext-real Element of REAL
L * N is Element of the carrier of T
the Mult of T is V16() V19(K7(REAL, the carrier of T)) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of T), the carrier of T))
K7(REAL, the carrier of T) is set
K7(K7(REAL, the carrier of T), the carrier of T) is set
K6(K7(K7(REAL, the carrier of T), the carrier of T)) is set
K249( the Mult of T,L,N) is set
R is Element of the carrier of T
s1 is V11() real ext-real Element of REAL
N2 is V11() real ext-real set
||.N.|| is V11() real ext-real Element of REAL
L is V11() real ext-real Element of REAL
||.N.|| + L is V11() real ext-real Element of REAL
0 + 0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
s1 / (||.N.|| + L) is V11() real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
R + x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
(T,f,S) . n is Element of the carrier of T
((T,f,S) . n) - R is Element of the carrier of T
- R is Element of the carrier of T
K192(T,((T,f,S) . n),(- R)) is Element of the carrier of T
||.(((T,f,S) . n) - R).|| is V11() real ext-real Element of REAL
S . n is V11() real ext-real Element of REAL
(S . n) - L is V11() real ext-real Element of REAL
abs ((S . n) - L) is V11() real ext-real Element of REAL
((S . n) - L) * N is Element of the carrier of T
K249( the Mult of T,((S . n) - L),N) is set
||.(((S . n) - L) * N).|| is V11() real ext-real Element of REAL
||.N.|| * (abs ((S . n) - L)) is V11() real ext-real Element of REAL
||.N.|| * (s1 / (||.N.|| + L)) is V11() real ext-real Element of REAL
f . n is Element of the carrier of T
(S . n) * (f . n) is Element of the carrier of T
K249( the Mult of T,(S . n),(f . n)) is set
((S . n) * (f . n)) - (L * N) is Element of the carrier of T
- (L * N) is Element of the carrier of T
K192(T,((S . n) * (f . n)),(- (L * N))) is Element of the carrier of T
||.(((S . n) * (f . n)) - (L * N)).|| is V11() real ext-real Element of REAL
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
((S . n) * (f . n)) - (0. T) is Element of the carrier of T
- (0. T) is Element of the carrier of T
K192(T,((S . n) * (f . n)),(- (0. T))) is Element of the carrier of T
(((S . n) * (f . n)) - (0. T)) - (L * N) is Element of the carrier of T
K192(T,(((S . n) * (f . n)) - (0. T)),(- (L * N))) is Element of the carrier of T
||.((((S . n) * (f . n)) - (0. T)) - (L * N)).|| is V11() real ext-real Element of REAL
(S . n) * N is Element of the carrier of T
K249( the Mult of T,(S . n),N) is set
((S . n) * N) - ((S . n) * N) is Element of the carrier of T
- ((S . n) * N) is Element of the carrier of T
K192(T,((S . n) * N),(- ((S . n) * N))) is Element of the carrier of T
((S . n) * (f . n)) - (((S . n) * N) - ((S . n) * N)) is Element of the carrier of T
- (((S . n) * N) - ((S . n) * N)) is Element of the carrier of T
K192(T,((S . n) * (f . n)),(- (((S . n) * N) - ((S . n) * N)))) is Element of the carrier of T
(((S . n) * (f . n)) - (((S . n) * N) - ((S . n) * N))) - (L * N) is Element of the carrier of T
K192(T,(((S . n) * (f . n)) - (((S . n) * N) - ((S . n) * N))),(- (L * N))) is Element of the carrier of T
||.((((S . n) * (f . n)) - (((S . n) * N) - ((S . n) * N))) - (L * N)).|| is V11() real ext-real Element of REAL
((S . n) * (f . n)) - ((S . n) * N) is Element of the carrier of T
K192(T,((S . n) * (f . n)),(- ((S . n) * N))) is Element of the carrier of T
(((S . n) * (f . n)) - ((S . n) * N)) + ((S . n) * N) is Element of the carrier of T
((((S . n) * (f . n)) - ((S . n) * N)) + ((S . n) * N)) - (L * N) is Element of the carrier of T
K192(T,((((S . n) * (f . n)) - ((S . n) * N)) + ((S . n) * N)),(- (L * N))) is Element of the carrier of T
||.(((((S . n) * (f . n)) - ((S . n) * N)) + ((S . n) * N)) - (L * N)).|| is V11() real ext-real Element of REAL
(f . n) - N is Element of the carrier of T
- N is Element of the carrier of T
K192(T,(f . n),(- N)) is Element of the carrier of T
(S . n) * ((f . n) - N) is Element of the carrier of T
K249( the Mult of T,(S . n),((f . n) - N)) is set
((S . n) * ((f . n) - N)) + ((S . n) * N) is Element of the carrier of T
(((S . n) * ((f . n) - N)) + ((S . n) * N)) - (L * N) is Element of the carrier of T
K192(T,(((S . n) * ((f . n) - N)) + ((S . n) * N)),(- (L * N))) is Element of the carrier of T
||.((((S . n) * ((f . n) - N)) + ((S . n) * N)) - (L * N)).|| is V11() real ext-real Element of REAL
((S . n) * N) - (L * N) is Element of the carrier of T
K192(T,((S . n) * N),(- (L * N))) is Element of the carrier of T
((S . n) * ((f . n) - N)) + (((S . n) * N) - (L * N)) is Element of the carrier of T
||.(((S . n) * ((f . n) - N)) + (((S . n) * N) - (L * N))).|| is V11() real ext-real Element of REAL
((S . n) * ((f . n) - N)) + (((S . n) - L) * N) is Element of the carrier of T
||.(((S . n) * ((f . n) - N)) + (((S . n) - L) * N)).|| is V11() real ext-real Element of REAL
||.((S . n) * ((f . n) - N)).|| is V11() real ext-real Element of REAL
||.((S . n) * ((f . n) - N)).|| + ||.(((S . n) - L) * N).|| is V11() real ext-real Element of REAL
||.((f . n) - N).|| is V11() real ext-real Element of REAL
abs (S . n) is V11() real ext-real Element of REAL
L * (s1 / (||.N.|| + L)) is V11() real ext-real Element of REAL
(abs (S . n)) * ||.((f . n) - N).|| is V11() real ext-real Element of REAL
(L * (s1 / (||.N.|| + L))) + (||.N.|| * (s1 / (||.N.|| + L))) is V11() real ext-real Element of REAL
(s1 / (||.N.|| + L)) * (||.N.|| + L) is V11() real ext-real Element of REAL
T is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() set
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
S is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
lim S is V11() real ext-real Element of REAL
f is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(T,f,S) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
lim (T,f,S) is Element of the carrier of T
lim f is Element of the carrier of T
(lim S) * (lim f) is Element of the carrier of T
the Mult of T is V16() V19(K7(REAL, the carrier of T)) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of T), the carrier of T))
K7(REAL, the carrier of T) is set
K7(K7(REAL, the carrier of T), the carrier of T) is set
K6(K7(K7(REAL, the carrier of T), the carrier of T)) is set
K249( the Mult of T,(lim S),(lim f)) is set
N is V11() real ext-real Element of REAL
N * (lim f) is Element of the carrier of T
K249( the Mult of T,N,(lim f)) is set
R is V11() real ext-real set
||.(lim f).|| is V11() real ext-real Element of REAL
s1 is V11() real ext-real Element of REAL
||.(lim f).|| + s1 is V11() real ext-real Element of REAL
0 + 0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
N2 is V11() real ext-real Element of REAL
N2 / (||.(lim f).|| + s1) is V11() real ext-real Element of REAL
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
R is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
L + R is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
(T,f,S) . y is Element of the carrier of T
((T,f,S) . y) - (N * (lim f)) is Element of the carrier of T
- (N * (lim f)) is Element of the carrier of T
K192(T,((T,f,S) . y),(- (N * (lim f)))) is Element of the carrier of T
||.(((T,f,S) . y) - (N * (lim f))).|| is V11() real ext-real Element of REAL
S . y is V11() real ext-real Element of REAL
(S . y) - N is V11() real ext-real Element of REAL
abs ((S . y) - N) is V11() real ext-real Element of REAL
((S . y) - N) * (lim f) is Element of the carrier of T
K249( the Mult of T,((S . y) - N),(lim f)) is set
||.(((S . y) - N) * (lim f)).|| is V11() real ext-real Element of REAL
||.(lim f).|| * (abs ((S . y) - N)) is V11() real ext-real Element of REAL
||.(lim f).|| * (N2 / (||.(lim f).|| + s1)) is V11() real ext-real Element of REAL
abs (S . y) is V11() real ext-real Element of REAL
f . y is Element of the carrier of T
(f . y) - (lim f) is Element of the carrier of T
- (lim f) is Element of the carrier of T
K192(T,(f . y),(- (lim f))) is Element of the carrier of T
||.((f . y) - (lim f)).|| is V11() real ext-real Element of REAL
(S . y) * (f . y) is Element of the carrier of T
K249( the Mult of T,(S . y),(f . y)) is set
((S . y) * (f . y)) - (N * (lim f)) is Element of the carrier of T
K192(T,((S . y) * (f . y)),(- (N * (lim f)))) is Element of the carrier of T
||.(((S . y) * (f . y)) - (N * (lim f))).|| is V11() real ext-real Element of REAL
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
((S . y) * (f . y)) - (0. T) is Element of the carrier of T
- (0. T) is Element of the carrier of T
K192(T,((S . y) * (f . y)),(- (0. T))) is Element of the carrier of T
(((S . y) * (f . y)) - (0. T)) - (N * (lim f)) is Element of the carrier of T
K192(T,(((S . y) * (f . y)) - (0. T)),(- (N * (lim f)))) is Element of the carrier of T
||.((((S . y) * (f . y)) - (0. T)) - (N * (lim f))).|| is V11() real ext-real Element of REAL
(S . y) * (lim f) is Element of the carrier of T
K249( the Mult of T,(S . y),(lim f)) is set
((S . y) * (lim f)) - ((S . y) * (lim f)) is Element of the carrier of T
- ((S . y) * (lim f)) is Element of the carrier of T
K192(T,((S . y) * (lim f)),(- ((S . y) * (lim f)))) is Element of the carrier of T
((S . y) * (f . y)) - (((S . y) * (lim f)) - ((S . y) * (lim f))) is Element of the carrier of T
- (((S . y) * (lim f)) - ((S . y) * (lim f))) is Element of the carrier of T
K192(T,((S . y) * (f . y)),(- (((S . y) * (lim f)) - ((S . y) * (lim f))))) is Element of the carrier of T
(((S . y) * (f . y)) - (((S . y) * (lim f)) - ((S . y) * (lim f)))) - (N * (lim f)) is Element of the carrier of T
K192(T,(((S . y) * (f . y)) - (((S . y) * (lim f)) - ((S . y) * (lim f)))),(- (N * (lim f)))) is Element of the carrier of T
||.((((S . y) * (f . y)) - (((S . y) * (lim f)) - ((S . y) * (lim f)))) - (N * (lim f))).|| is V11() real ext-real Element of REAL
((S . y) * (f . y)) - ((S . y) * (lim f)) is Element of the carrier of T
K192(T,((S . y) * (f . y)),(- ((S . y) * (lim f)))) is Element of the carrier of T
(((S . y) * (f . y)) - ((S . y) * (lim f))) + ((S . y) * (lim f)) is Element of the carrier of T
((((S . y) * (f . y)) - ((S . y) * (lim f))) + ((S . y) * (lim f))) - (N * (lim f)) is Element of the carrier of T
K192(T,((((S . y) * (f . y)) - ((S . y) * (lim f))) + ((S . y) * (lim f))),(- (N * (lim f)))) is Element of the carrier of T
||.(((((S . y) * (f . y)) - ((S . y) * (lim f))) + ((S . y) * (lim f))) - (N * (lim f))).|| is V11() real ext-real Element of REAL
(S . y) * ((f . y) - (lim f)) is Element of the carrier of T
K249( the Mult of T,(S . y),((f . y) - (lim f))) is set
((S . y) * ((f . y) - (lim f))) + ((S . y) * (lim f)) is Element of the carrier of T
(((S . y) * ((f . y) - (lim f))) + ((S . y) * (lim f))) - (N * (lim f)) is Element of the carrier of T
K192(T,(((S . y) * ((f . y) - (lim f))) + ((S . y) * (lim f))),(- (N * (lim f)))) is Element of the carrier of T
||.((((S . y) * ((f . y) - (lim f))) + ((S . y) * (lim f))) - (N * (lim f))).|| is V11() real ext-real Element of REAL
((S . y) * (lim f)) - (N * (lim f)) is Element of the carrier of T
K192(T,((S . y) * (lim f)),(- (N * (lim f)))) is Element of the carrier of T
((S . y) * ((f . y) - (lim f))) + (((S . y) * (lim f)) - (N * (lim f))) is Element of the carrier of T
||.(((S . y) * ((f . y) - (lim f))) + (((S . y) * (lim f)) - (N * (lim f)))).|| is V11() real ext-real Element of REAL
((S . y) * ((f . y) - (lim f))) + (((S . y) - N) * (lim f)) is Element of the carrier of T
||.(((S . y) * ((f . y) - (lim f))) + (((S . y) - N) * (lim f))).|| is V11() real ext-real Element of REAL
||.((S . y) * ((f . y) - (lim f))).|| is V11() real ext-real Element of REAL
||.((S . y) * ((f . y) - (lim f))).|| + ||.(((S . y) - N) * (lim f)).|| is V11() real ext-real Element of REAL
s1 * (N2 / (||.(lim f).|| + s1)) is V11() real ext-real Element of REAL
(abs (S . y)) * ||.((f . y) - (lim f)).|| is V11() real ext-real Element of REAL
(s1 * (N2 / (||.(lim f).|| + s1))) + (||.(lim f).|| * (N2 / (||.(lim f).|| + s1))) is V11() real ext-real Element of REAL
(N2 / (||.(lim f).|| + s1)) * (||.(lim f).|| + s1) is V11() real ext-real Element of REAL
T is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
S is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() set
K7(NAT, the carrier of S) is set
K6(K7(NAT, the carrier of S)) is set
f is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
f ^\ T is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total subsequence of f
x0 is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
f + x0 is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
(f + x0) ^\ T is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total subsequence of f + x0
x0 ^\ T is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total subsequence of x0
(f ^\ T) + (x0 ^\ T) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
N is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
((f + x0) ^\ T) . N is Element of the carrier of S
N + T is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
(f + x0) . (N + T) is Element of the carrier of S
f . (N + T) is Element of the carrier of S
x0 . (N + T) is Element of the carrier of S
(f . (N + T)) + (x0 . (N + T)) is Element of the carrier of S
(f ^\ T) . N is Element of the carrier of S
((f ^\ T) . N) + (x0 . (N + T)) is Element of the carrier of S
(x0 ^\ T) . N is Element of the carrier of S
((f ^\ T) . N) + ((x0 ^\ T) . N) is Element of the carrier of S
((f ^\ T) + (x0 ^\ T)) . N is Element of the carrier of S
T is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
S is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() set
K7(NAT, the carrier of S) is set
K6(K7(NAT, the carrier of S)) is set
f is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
f ^\ T is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total subsequence of f
x0 is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
f - x0 is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
(f - x0) ^\ T is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total subsequence of f - x0
x0 ^\ T is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total subsequence of x0
(f ^\ T) - (x0 ^\ T) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
N is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
((f - x0) ^\ T) . N is Element of the carrier of S
N + T is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
(f - x0) . (N + T) is Element of the carrier of S
f . (N + T) is Element of the carrier of S
x0 . (N + T) is Element of the carrier of S
(f . (N + T)) - (x0 . (N + T)) is Element of the carrier of S
- (x0 . (N + T)) is Element of the carrier of S
K192(S,(f . (N + T)),(- (x0 . (N + T)))) is Element of the carrier of S
(f ^\ T) . N is Element of the carrier of S
((f ^\ T) . N) - (x0 . (N + T)) is Element of the carrier of S
K192(S,((f ^\ T) . N),(- (x0 . (N + T)))) is Element of the carrier of S
(x0 ^\ T) . N is Element of the carrier of S
((f ^\ T) . N) - ((x0 ^\ T) . N) is Element of the carrier of S
- ((x0 ^\ T) . N) is Element of the carrier of S
K192(S,((f ^\ T) . N),(- ((x0 ^\ T) . N))) is Element of the carrier of S
((f ^\ T) - (x0 ^\ T)) . N is Element of the carrier of S
T is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
S is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() set
K7(NAT, the carrier of S) is set
K6(K7(NAT, the carrier of S)) is set
f is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
f ^\ T is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total subsequence of f
x0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
(f ^\ T) . x0 is Element of the carrier of S
x0 + T is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
f . (x0 + T) is Element of the carrier of S
0. S is V68(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
T is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() set
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
T is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() set
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
S is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
lim S is Element of the carrier of T
f is Element of the carrier of T
N is V11() real ext-real Element of REAL
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
R is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
S . R is Element of the carrier of T
x0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
S . x0 is Element of the carrier of T
(S . R) - (S . x0) is Element of the carrier of T
- (S . x0) is Element of the carrier of T
K192(T,(S . R),(- (S . x0))) is Element of the carrier of T
||.((S . R) - (S . x0)).|| is V11() real ext-real Element of REAL
f - (S . x0) is Element of the carrier of T
K192(T,f,(- (S . x0))) is Element of the carrier of T
||.(f - (S . x0)).|| is V11() real ext-real Element of REAL
f - f is Element of the carrier of T
- f is Element of the carrier of T
K192(T,f,(- f)) is Element of the carrier of T
||.(f - f).|| is V11() real ext-real Element of REAL
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
||.(0. T).|| is V11() real ext-real Element of REAL
x0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
S . x0 is Element of the carrier of T
T is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() set
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
S is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
f is Element of the carrier of T
x0 is V11() real ext-real Element of REAL
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
N is V68(T) Element of the carrier of T
L is V11() real ext-real Element of REAL
||.f.|| is V11() real ext-real Element of REAL
||.f.|| + 1 is V11() real ext-real Element of REAL
L / (||.f.|| + 1) is V11() real ext-real Element of REAL
R is V11() real ext-real Element of REAL
R * ||.f.|| is V11() real ext-real Element of REAL
||.f.|| + 0 is V11() real ext-real Element of REAL
0 + 0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
R * (||.f.|| + 1) is V11() real ext-real Element of REAL
R is V11() real ext-real Element of REAL
R * ||.f.|| is V11() real ext-real Element of REAL
R is V11() real ext-real Element of REAL
R * ||.f.|| is V11() real ext-real Element of REAL
R " is V11() real ext-real Element of REAL
s1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
s1 + x0 is V11() real ext-real Element of REAL
(R ") + 0 is V11() real ext-real Element of REAL
1 / (R ") is V11() real ext-real Element of REAL
1 / (s1 + x0) is V11() real ext-real Element of REAL
(R ") " is V11() real ext-real Element of REAL
1 * ((R ") ") is V11() real ext-real Element of REAL
N2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
S . L is Element of the carrier of T
(S . L) - N is Element of the carrier of T
- N is Element of the carrier of T
K192(T,(S . L),(- N)) is Element of the carrier of T
||.((S . L) - N).|| is V11() real ext-real Element of REAL
N2 + x0 is V11() real ext-real Element of REAL
L + x0 is V11() real ext-real Element of REAL
1 / (L + x0) is V11() real ext-real Element of REAL
1 / (N2 + x0) is V11() real ext-real Element of REAL
(1 / (L + x0)) * ||.f.|| is V11() real ext-real Element of REAL
(1 / (L + x0)) * f is Element of the carrier of T
the Mult of T is V16() V19(K7(REAL, the carrier of T)) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of T), the carrier of T))
K7(REAL, the carrier of T) is set
K7(K7(REAL, the carrier of T), the carrier of T) is set
K6(K7(K7(REAL, the carrier of T), the carrier of T)) is set
K249( the Mult of T,(1 / (L + x0)),f) is set
((1 / (L + x0)) * f) - (0. T) is Element of the carrier of T
- (0. T) is Element of the carrier of T
K192(T,((1 / (L + x0)) * f),(- (0. T))) is Element of the carrier of T
||.(((1 / (L + x0)) * f) - (0. T)).|| is V11() real ext-real Element of REAL
||.((1 / (L + x0)) * f).|| is V11() real ext-real Element of REAL
abs (1 / (L + x0)) is V11() real ext-real Element of REAL
(abs (1 / (L + x0))) * ||.f.|| is V11() real ext-real Element of REAL
T is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() set
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
S is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
lim S is Element of the carrier of T
f is Element of the carrier of T
x0 is V11() real ext-real Element of REAL
||.f.|| is V11() real ext-real Element of REAL
N is V11() real ext-real Element of REAL
||.f.|| + 1 is V11() real ext-real Element of REAL
N / (||.f.|| + 1) is V11() real ext-real Element of REAL
L is V11() real ext-real Element of REAL
L * ||.f.|| is V11() real ext-real Element of REAL
||.f.|| + 0 is V11() real ext-real Element of REAL
0 + 0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
L * (||.f.|| + 1) is V11() real ext-real Element of REAL
L is V11() real ext-real Element of REAL
L * ||.f.|| is V11() real ext-real Element of REAL
L is V11() real ext-real Element of REAL
L * ||.f.|| is V11() real ext-real Element of REAL
L " is V11() real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
R + x0 is V11() real ext-real Element of REAL
(L ") + 0 is V11() real ext-real Element of REAL
1 / (L ") is V11() real ext-real Element of REAL
1 / (R + x0) is V11() real ext-real Element of REAL
(L ") " is V11() real ext-real Element of REAL
1 * ((L ") ") is V11() real ext-real Element of REAL
s1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
N2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
s1 + x0 is V11() real ext-real Element of REAL
N2 + x0 is V11() real ext-real Element of REAL
1 / (N2 + x0) is V11() real ext-real Element of REAL
1 / (s1 + x0) is V11() real ext-real Element of REAL
(1 / (N2 + x0)) * ||.f.|| is V11() real ext-real Element of REAL
S . N2 is Element of the carrier of T
(S . N2) - (0. T) is Element of the carrier of T
- (0. T) is Element of the carrier of T
K192(T,(S . N2),(- (0. T))) is Element of the carrier of T
||.((S . N2) - (0. T)).|| is V11() real ext-real Element of REAL
(1 / (N2 + x0)) * f is Element of the carrier of T
the Mult of T is V16() V19(K7(REAL, the carrier of T)) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of T), the carrier of T))
K7(REAL, the carrier of T) is set
K7(K7(REAL, the carrier of T), the carrier of T) is set
K6(K7(K7(REAL, the carrier of T), the carrier of T)) is set
K249( the Mult of T,(1 / (N2 + x0)),f) is set
((1 / (N2 + x0)) * f) - (0. T) is Element of the carrier of T
K192(T,((1 / (N2 + x0)) * f),(- (0. T))) is Element of the carrier of T
||.(((1 / (N2 + x0)) * f) - (0. T)).|| is V11() real ext-real Element of REAL
||.((1 / (N2 + x0)) * f).|| is V11() real ext-real Element of REAL
abs (1 / (N2 + x0)) is V11() real ext-real Element of REAL
(abs (1 / (N2 + x0))) * ||.f.|| is V11() real ext-real Element of REAL
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
S is Element of the carrier of T
f is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
x0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
x0 + 1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
(x0 + 1) " is V11() real ext-real Element of REAL
1 / (x0 + 1) is V11() real ext-real Element of REAL
f . x0 is Element of the carrier of T
(1 / (x0 + 1)) * S is Element of the carrier of T
the Mult of T is V16() V19(K7(REAL, the carrier of T)) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of T), the carrier of T))
K7(REAL, the carrier of T) is set
K7(K7(REAL, the carrier of T), the carrier of T) is set
K6(K7(K7(REAL, the carrier of T), the carrier of T)) is set
K249( the Mult of T,(1 / (x0 + 1)),S) is set
lim f is Element of the carrier of T
T is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() set
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
S is V1() V16() non-empty V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K6(K7(NAT,REAL))
f is Element of the carrier of T
(T,f,S) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
x0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
S . x0 is V11() real ext-real Element of REAL
(T,f,S) . x0 is Element of the carrier of T
(S . x0) * f is Element of the carrier of T
the Mult of T is V16() V19(K7(REAL, the carrier of T)) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of T), the carrier of T))
K7(REAL, the carrier of T) is set
K7(K7(REAL, the carrier of T), the carrier of T) is set
K6(K7(K7(REAL, the carrier of T), the carrier of T)) is set
K249( the Mult of T,(S . x0),f) is set
x0 is V11() real ext-real Element of REAL
||.f.|| is V11() real ext-real Element of REAL
||.f.|| + 1 is V11() real ext-real Element of REAL
x0 / (||.f.|| + 1) is V11() real ext-real Element of REAL
N is V11() real ext-real Element of REAL
N * ||.f.|| is V11() real ext-real Element of REAL
||.f.|| + 0 is V11() real ext-real Element of REAL
0 + 0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
N * (||.f.|| + 1) is V11() real ext-real Element of REAL
N is V11() real ext-real Element of REAL
N * ||.f.|| is V11() real ext-real Element of REAL
N is V11() real ext-real Element of REAL
N * ||.f.|| is V11() real ext-real Element of REAL
lim S is V1() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V170() V171() V172() V173() V174() V175() V176() Element of REAL
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
R is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
s1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
S . s1 is V11() real ext-real Element of REAL
(S . s1) - 0 is V11() real ext-real Element of REAL
abs ((S . s1) - 0) is V11() real ext-real Element of REAL
(T,f,S) . s1 is Element of the carrier of T
((T,f,S) . s1) - (0. T) is Element of the carrier of T
- (0. T) is Element of the carrier of T
K192(T,((T,f,S) . s1),(- (0. T))) is Element of the carrier of T
||.(((T,f,S) . s1) - (0. T)).|| is V11() real ext-real Element of REAL
(S . s1) * f is Element of the carrier of T
K249( the Mult of T,(S . s1),f) is set
((S . s1) * f) - (0. T) is Element of the carrier of T
K192(T,((S . s1) * f),(- (0. T))) is Element of the carrier of T
||.(((S . s1) * f) - (0. T)).|| is V11() real ext-real Element of REAL
||.((S . s1) * f).|| is V11() real ext-real Element of REAL
abs (S . s1) is V11() real ext-real Element of REAL
(abs (S . s1)) * ||.f.|| is V11() real ext-real Element of REAL
lim (T,f,S) is Element of the carrier of T
T is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() set
K6( the carrier of T) is set
S is Element of K6( the carrier of T)
f is set
x0 is set
f is Element of the carrier of T
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
NAT --> (0. T) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like constant total quasi_total Element of K6(K7(NAT, the carrier of T))
S is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
K7( the carrier of T, the carrier of S) is set
K6(K7( the carrier of T, the carrier of S)) is set
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
K7( the carrier of T, the carrier of S) is set
K6(K7( the carrier of T, the carrier of S)) is set
0. S is V68(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
the carrier of T --> (0. S) is V1() V16() V19( the carrier of T) V20( the carrier of S) Function-like constant total quasi_total Element of K6(K7( the carrier of T, the carrier of S))
f is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
dom f is Element of K6( the carrier of T)
K6( the carrier of T) is set
x0 is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total non-zero (T, 0. T) Element of K6(K7(NAT, the carrier of T))
N is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
x0 . N is Element of the carrier of T
f /. (x0 . N) is Element of the carrier of S
f . (x0 . N) is set
rng x0 is Element of K6( the carrier of T)
f /* x0 is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
K7(NAT, the carrier of S) is set
K6(K7(NAT, the carrier of S)) is set
||.x0.|| is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
||.x0.|| " is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(S,(f /* x0),(||.x0.|| ")) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
(S,(f /* x0),(||.x0.|| ")) . N is Element of the carrier of S
(f /* x0) . N is Element of the carrier of S
(||.x0.|| ") . N is V11() real ext-real Element of REAL
((||.x0.|| ") . N) * ((f /* x0) . N) is Element of the carrier of S
the Mult of S is V16() V19(K7(REAL, the carrier of S)) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of S), the carrier of S))
K7(REAL, the carrier of S) is set
K7(K7(REAL, the carrier of S), the carrier of S) is set
K6(K7(K7(REAL, the carrier of S), the carrier of S)) is set
K249( the Mult of S,((||.x0.|| ") . N),((f /* x0) . N)) is set
((||.x0.|| ") . N) * (f /. (x0 . N)) is Element of the carrier of S
K249( the Mult of S,((||.x0.|| ") . N),(f /. (x0 . N))) is set
(S,(f /* x0),(||.x0.|| ")) . 0 is Element of the carrier of S
lim (S,(f /* x0),(||.x0.|| ")) is Element of the carrier of S
x0 is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total non-zero (T, 0. T) Element of K6(K7(NAT, the carrier of T))
f /* x0 is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
||.x0.|| is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
||.x0.|| " is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(S,(f /* x0),(||.x0.|| ")) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
N is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total non-zero (T, 0. T) Element of K6(K7(NAT, the carrier of T))
f /* N is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
||.N.|| is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
||.N.|| " is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(S,(f /* N),(||.N.|| ")) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
lim (S,(f /* N),(||.N.|| ")) is Element of the carrier of S
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
K7( the carrier of T, the carrier of S) is set
K6(K7( the carrier of T, the carrier of S)) is set
f is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
x0 is V11() real ext-real Element of REAL
x0 is V11() real ext-real Element of REAL
N is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
N + 1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
1 / (N + 1) is V11() real ext-real Element of REAL
(N + 1) " is V11() real ext-real Element of REAL
1 * ((N + 1) ") is V11() real ext-real Element of REAL
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
N is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
L is V11() real ext-real Element of REAL
L " is V11() real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
R + 1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
(L ") + 0 is V11() real ext-real Element of REAL
1 / (L ") is V11() real ext-real Element of REAL
1 / (R + 1) is V11() real ext-real Element of REAL
s1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
N2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
s1 + 1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
N2 + 1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
1 / (N2 + 1) is V11() real ext-real Element of REAL
N . N2 is Element of the carrier of T
||.(N . N2).|| is V11() real ext-real Element of REAL
(N . N2) - (0. T) is Element of the carrier of T
- (0. T) is Element of the carrier of T
K192(T,(N . N2),(- (0. T))) is Element of the carrier of T
||.((N . N2) - (0. T)).|| is V11() real ext-real Element of REAL
1 / (s1 + 1) is V11() real ext-real Element of REAL
lim N is Element of the carrier of T
L is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total non-zero (T, 0. T) Element of K6(K7(NAT, the carrier of T))
f /* L is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
K7(NAT, the carrier of S) is set
K6(K7(NAT, the carrier of S)) is set
||.L.|| is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
||.L.|| " is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(S,(f /* L),(||.L.|| ")) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
lim (S,(f /* L),(||.L.|| ")) is Element of the carrier of S
0. S is V68(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
R is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
(S,(f /* L),(||.L.|| ")) . R is Element of the carrier of S
((S,(f /* L),(||.L.|| ")) . R) - (0. S) is Element of the carrier of S
- (0. S) is Element of the carrier of S
K192(S,((S,(f /* L),(||.L.|| ")) . R),(- (0. S))) is Element of the carrier of S
||.(((S,(f /* L),(||.L.|| ")) . R) - (0. S)).|| is V11() real ext-real Element of REAL
L . R is Element of the carrier of T
||.(L . R).|| is V11() real ext-real Element of REAL
f /. (L . R) is Element of the carrier of S
||.(L . R).|| " is V11() real ext-real Element of REAL
(||.(L . R).|| ") * (f /. (L . R)) is Element of the carrier of S
the Mult of S is V16() V19(K7(REAL, the carrier of S)) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of S), the carrier of S))
K7(REAL, the carrier of S) is set
K7(K7(REAL, the carrier of S), the carrier of S) is set
K6(K7(K7(REAL, the carrier of S), the carrier of S)) is set
K249( the Mult of S,(||.(L . R).|| "),(f /. (L . R))) is set
||.((||.(L . R).|| ") * (f /. (L . R))).|| is V11() real ext-real Element of REAL
abs (||.(L . R).|| ") is V11() real ext-real Element of REAL
||.(f /. (L . R)).|| is V11() real ext-real Element of REAL
(abs (||.(L . R).|| ")) * ||.(f /. (L . R)).|| is V11() real ext-real Element of REAL
(||.(L . R).|| ") * ||.(f /. (L . R)).|| is V11() real ext-real Element of REAL
dom f is Element of K6( the carrier of T)
K6( the carrier of T) is set
rng L is Element of K6( the carrier of T)
||.((S,(f /* L),(||.L.|| ")) . R).|| is V11() real ext-real Element of REAL
(f /* L) . R is Element of the carrier of S
(||.L.|| ") . R is V11() real ext-real Element of REAL
((||.L.|| ") . R) * ((f /* L) . R) is Element of the carrier of S
K249( the Mult of S,((||.L.|| ") . R),((f /* L) . R)) is set
||.(((||.L.|| ") . R) * ((f /* L) . R)).|| is V11() real ext-real Element of REAL
||.L.|| . R is V11() real ext-real Element of REAL
(||.L.|| . R) " is V11() real ext-real Element of REAL
((||.L.|| . R) ") * ((f /* L) . R) is Element of the carrier of S
K249( the Mult of S,((||.L.|| . R) "),((f /* L) . R)) is set
||.(((||.L.|| . R) ") * ((f /* L) . R)).|| is V11() real ext-real Element of REAL
(||.(L . R).|| ") * ((f /* L) . R) is Element of the carrier of S
K249( the Mult of S,(||.(L . R).|| "),((f /* L) . R)) is set
||.((||.(L . R).|| ") * ((f /* L) . R)).|| is V11() real ext-real Element of REAL
s1 is V11() real ext-real Element of REAL
x0 is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total non-zero (T, 0. T) Element of K6(K7(NAT, the carrier of T))
lim x0 is Element of the carrier of T
N is V11() real ext-real Element of REAL
L is V11() real ext-real Element of REAL
R is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
s1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
f /* x0 is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
||.x0.|| is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
||.x0.|| " is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(S,(f /* x0),(||.x0.|| ")) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
rng x0 is Element of K6( the carrier of T)
N2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
(S,(f /* x0),(||.x0.|| ")) . N2 is Element of the carrier of S
((S,(f /* x0),(||.x0.|| ")) . N2) - (0. S) is Element of the carrier of S
K192(S,((S,(f /* x0),(||.x0.|| ")) . N2),(- (0. S))) is Element of the carrier of S
||.(((S,(f /* x0),(||.x0.|| ")) . N2) - (0. S)).|| is V11() real ext-real Element of REAL
x0 . N2 is Element of the carrier of T
(x0 . N2) - (0. T) is Element of the carrier of T
K192(T,(x0 . N2),(- (0. T))) is Element of the carrier of T
||.((x0 . N2) - (0. T)).|| is V11() real ext-real Element of REAL
||.(x0 . N2).|| is V11() real ext-real Element of REAL
||.(x0 . N2).|| " is V11() real ext-real Element of REAL
f /. (x0 . N2) is Element of the carrier of S
||.(f /. (x0 . N2)).|| is V11() real ext-real Element of REAL
(||.(x0 . N2).|| ") * ||.(f /. (x0 . N2)).|| is V11() real ext-real Element of REAL
abs (||.(x0 . N2).|| ") is V11() real ext-real Element of REAL
(abs (||.(x0 . N2).|| ")) * ||.(f /. (x0 . N2)).|| is V11() real ext-real Element of REAL
(||.(x0 . N2).|| ") * (f /. (x0 . N2)) is Element of the carrier of S
K249( the Mult of S,(||.(x0 . N2).|| "),(f /. (x0 . N2))) is set
||.((||.(x0 . N2).|| ") * (f /. (x0 . N2))).|| is V11() real ext-real Element of REAL
(f /* x0) . N2 is Element of the carrier of S
(||.(x0 . N2).|| ") * ((f /* x0) . N2) is Element of the carrier of S
K249( the Mult of S,(||.(x0 . N2).|| "),((f /* x0) . N2)) is set
||.((||.(x0 . N2).|| ") * ((f /* x0) . N2)).|| is V11() real ext-real Element of REAL
||.x0.|| . N2 is V11() real ext-real Element of REAL
(||.x0.|| . N2) " is V11() real ext-real Element of REAL
((||.x0.|| . N2) ") * ((f /* x0) . N2) is Element of the carrier of S
K249( the Mult of S,((||.x0.|| . N2) "),((f /* x0) . N2)) is set
||.(((||.x0.|| . N2) ") * ((f /* x0) . N2)).|| is V11() real ext-real Element of REAL
(||.x0.|| ") . N2 is V11() real ext-real Element of REAL
((||.x0.|| ") . N2) * ((f /* x0) . N2) is Element of the carrier of S
K249( the Mult of S,((||.x0.|| ") . N2),((f /* x0) . N2)) is set
||.(((||.x0.|| ") . N2) * ((f /* x0) . N2)).|| is V11() real ext-real Element of REAL
||.((S,(f /* x0),(||.x0.|| ")) . N2).|| is V11() real ext-real Element of REAL
lim (S,(f /* x0),(||.x0.|| ")) is Element of the carrier of S
x0 is V11() real ext-real Element of REAL
N is V11() real ext-real Element of REAL
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
K7( the carrier of T, the carrier of S) is set
K6(K7( the carrier of T, the carrier of S)) is set
0. S is V68(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
f is V16() V19( the carrier of T) V20( the carrier of S) Function-like (T,S) Element of K6(K7( the carrier of T, the carrier of S))
x0 is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total non-zero (T, 0. T) Element of K6(K7(NAT, the carrier of T))
f /* x0 is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
K7(NAT, the carrier of S) is set
K6(K7(NAT, the carrier of S)) is set
lim (f /* x0) is Element of the carrier of S
||.x0.|| is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
||.x0.|| " is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(S,(f /* x0),(||.x0.|| ")) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
N is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
x0 . N is Element of the carrier of T
||.(x0 . N).|| is V11() real ext-real Element of REAL
(S,(S,(f /* x0),(||.x0.|| ")),||.x0.||) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
(S,(S,(f /* x0),(||.x0.|| ")),||.x0.||) . N is Element of the carrier of S
(S,(f /* x0),(||.x0.|| ")) . N is Element of the carrier of S
||.x0.|| . N is V11() real ext-real Element of REAL
(||.x0.|| . N) * ((S,(f /* x0),(||.x0.|| ")) . N) is Element of the carrier of S
the Mult of S is V16() V19(K7(REAL, the carrier of S)) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of S), the carrier of S))
K7(REAL, the carrier of S) is set
K7(K7(REAL, the carrier of S), the carrier of S) is set
K6(K7(K7(REAL, the carrier of S), the carrier of S)) is set
K249( the Mult of S,(||.x0.|| . N),((S,(f /* x0),(||.x0.|| ")) . N)) is set
(f /* x0) . N is Element of the carrier of S
(||.x0.|| ") . N is V11() real ext-real Element of REAL
((||.x0.|| ") . N) * ((f /* x0) . N) is Element of the carrier of S
K249( the Mult of S,((||.x0.|| ") . N),((f /* x0) . N)) is set
(||.x0.|| . N) * (((||.x0.|| ") . N) * ((f /* x0) . N)) is Element of the carrier of S
K249( the Mult of S,(||.x0.|| . N),(((||.x0.|| ") . N) * ((f /* x0) . N))) is set
||.(x0 . N).|| * (((||.x0.|| ") . N) * ((f /* x0) . N)) is Element of the carrier of S
K249( the Mult of S,||.(x0 . N).||,(((||.x0.|| ") . N) * ((f /* x0) . N))) is set
(||.x0.|| . N) " is V11() real ext-real Element of REAL
((||.x0.|| . N) ") * ((f /* x0) . N) is Element of the carrier of S
K249( the Mult of S,((||.x0.|| . N) "),((f /* x0) . N)) is set
||.(x0 . N).|| * (((||.x0.|| . N) ") * ((f /* x0) . N)) is Element of the carrier of S
K249( the Mult of S,||.(x0 . N).||,(((||.x0.|| . N) ") * ((f /* x0) . N))) is set
||.(x0 . N).|| " is V11() real ext-real Element of REAL
(||.(x0 . N).|| ") * ((f /* x0) . N) is Element of the carrier of S
K249( the Mult of S,(||.(x0 . N).|| "),((f /* x0) . N)) is set
||.(x0 . N).|| * ((||.(x0 . N).|| ") * ((f /* x0) . N)) is Element of the carrier of S
K249( the Mult of S,||.(x0 . N).||,((||.(x0 . N).|| ") * ((f /* x0) . N))) is set
||.(x0 . N).|| * (||.(x0 . N).|| ") is V11() real ext-real Element of REAL
(||.(x0 . N).|| * (||.(x0 . N).|| ")) * ((f /* x0) . N) is Element of the carrier of S
K249( the Mult of S,(||.(x0 . N).|| * (||.(x0 . N).|| ")),((f /* x0) . N)) is set
1 * ((f /* x0) . N) is Element of the carrier of S
K249( the Mult of S,1,((f /* x0) . N)) is set
lim x0 is Element of the carrier of T
lim ||.x0.|| is V11() real ext-real Element of REAL
||.(0. T).|| is V11() real ext-real Element of REAL
lim (S,(f /* x0),(||.x0.|| ")) is Element of the carrier of S
0 * (0. S) is Element of the carrier of S
K249( the Mult of S,0,(0. S)) is set
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
K7( the carrier of T, the carrier of S) is set
K6(K7( the carrier of T, the carrier of S)) is set
f is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
x0 is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
f + x0 is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
f - x0 is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
N is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(f + x0) /* N is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
K7(NAT, the carrier of S) is set
K6(K7(NAT, the carrier of S)) is set
f /* N is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
x0 /* N is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
(f /* N) + (x0 /* N) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
(f - x0) /* N is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
(f /* N) - (x0 /* N) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
dom (f + x0) is Element of K6( the carrier of T)
K6( the carrier of T) is set
dom f is Element of K6( the carrier of T)
dom x0 is Element of K6( the carrier of T)
(dom f) /\ (dom x0) is Element of K6( the carrier of T)
rng N is Element of K6( the carrier of T)
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
K7( the carrier of T, the carrier of S) is set
K6(K7( the carrier of T, the carrier of S)) is set
f is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
x0 is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
f /* x0 is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
K7(NAT, the carrier of S) is set
K6(K7(NAT, the carrier of S)) is set
N is V11() real ext-real Element of REAL
N (#) f is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
(N (#) f) /* x0 is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
N * (f /* x0) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
dom f is Element of K6( the carrier of T)
K6( the carrier of T) is set
rng x0 is Element of K6( the carrier of T)
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
K7( the carrier of S, the carrier of T) is set
K6(K7( the carrier of S, the carrier of T)) is set
K7(NAT, the carrier of S) is set
K6(K7(NAT, the carrier of S)) is set
f is V16() V19( the carrier of S) V20( the carrier of T) Function-like Element of K6(K7( the carrier of S, the carrier of T))
dom f is Element of K6( the carrier of S)
K6( the carrier of S) is set
x0 is Element of the carrier of S
f /. x0 is Element of the carrier of T
N is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
rng N is Element of K6( the carrier of S)
lim N is Element of the carrier of S
f /* N is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
lim (f /* N) is Element of the carrier of T
N is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
rng N is Element of K6( the carrier of S)
lim N is Element of the carrier of S
f /* N is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
lim (f /* N) is Element of the carrier of T
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
N ^\ L is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total subsequence of N
R is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
(N ^\ L) . R is Element of the carrier of S
R + L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
N . (R + L) is Element of the carrier of S
rng (N ^\ L) is Element of K6( the carrier of S)
R is V11() real ext-real Element of REAL
s1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
N2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
f /* (N ^\ L) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(f /* (N ^\ L)) . N2 is Element of the carrier of T
((f /* (N ^\ L)) . N2) - (f /. x0) is Element of the carrier of T
- (f /. x0) is Element of the carrier of T
K192(T,((f /* (N ^\ L)) . N2),(- (f /. x0))) is Element of the carrier of T
||.(((f /* (N ^\ L)) . N2) - (f /. x0)).|| is V11() real ext-real Element of REAL
(N ^\ L) . N2 is Element of the carrier of S
f /. ((N ^\ L) . N2) is Element of the carrier of T
(f /. ((N ^\ L) . N2)) - (f /. x0) is Element of the carrier of T
K192(T,(f /. ((N ^\ L) . N2)),(- (f /. x0))) is Element of the carrier of T
||.((f /. ((N ^\ L) . N2)) - (f /. x0)).|| is V11() real ext-real Element of REAL
(f /. x0) - (f /. x0) is Element of the carrier of T
K192(T,(f /. x0),(- (f /. x0))) is Element of the carrier of T
||.((f /. x0) - (f /. x0)).|| is V11() real ext-real Element of REAL
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
||.(0. T).|| is V11() real ext-real Element of REAL
(f /* N) ^\ L is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total subsequence of f /* N
lim ((f /* N) ^\ L) is Element of the carrier of T
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
N . L is Element of the carrier of S
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
N . L is Element of the carrier of S
s1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
s1 + 1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
N2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
N . N2 is Element of the carrier of S
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
N . L is Element of the carrier of S
s1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
N2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
N . L is Element of the carrier of S
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
N . L is Element of the carrier of S
R is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
N . y is Element of the carrier of S
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
N . n is Element of the carrier of S
K6(K7(NAT,NAT)) is set
R is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
s1 is V1() V16() V19( NAT ) V20( NAT ) Function-like total quasi_total complex-valued ext-real-valued real-valued natural-valued Element of K6(K7(NAT,NAT))
s1 . 0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
rng s1 is V170() V171() V172() V173() V174() V175() Element of K6(NAT)
dom s1 is V170() V171() V172() V173() V174() V175() Element of K6(NAT)
N2 is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
N2 . L is V11() real ext-real Element of REAL
rng N2 is V170() V171() V172() Element of K6(REAL)
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
N2 . L is V11() real ext-real Element of REAL
L + 1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
N2 . (L + 1) is V11() real ext-real Element of REAL
L is V1() V16() V19( NAT ) V20( NAT ) Function-like total quasi_total complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing Element of K6(K7(NAT,NAT))
N * L is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total subsequence of N
lim (N * L) is Element of the carrier of S
R is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
N . R is Element of the carrier of S
R is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
N . R is Element of the carrier of S
L . 0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
N . x is Element of the carrier of S
y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
L . y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
N . x is Element of the carrier of S
y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
N . y is Element of the carrier of S
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
L . n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
y1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
L . y1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
L . n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
L . y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
L . y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
y + 1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
L . (y + 1) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
N . (L . (y + 1)) is Element of the carrier of S
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
L . n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
R is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
(N * L) . R is Element of the carrier of S
R + 1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
(N * L) . (R + 1) is Element of the carrier of S
L . R is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
L . (R + 1) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
N . y is Element of the carrier of S
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
N . n is Element of the carrier of S
N . (L . (R + 1)) is Element of the carrier of S
(N * L) . 0 is Element of the carrier of S
rng (N * L) is Element of K6( the carrier of S)
f /* (N * L) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
lim (f /* (N * L)) is Element of the carrier of T
R is V11() real ext-real Element of REAL
x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
L . x is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
N . n is Element of the carrier of S
(f /* N) . n is Element of the carrier of T
((f /* N) . n) - (f /. x0) is Element of the carrier of T
- (f /. x0) is Element of the carrier of T
K192(T,((f /* N) . n),(- (f /. x0))) is Element of the carrier of T
||.(((f /* N) . n) - (f /. x0)).|| is V11() real ext-real Element of REAL
f /. (N . n) is Element of the carrier of T
(f /. (N . n)) - (f /. x0) is Element of the carrier of T
K192(T,(f /. (N . n)),(- (f /. x0))) is Element of the carrier of T
||.((f /. (N . n)) - (f /. x0)).|| is V11() real ext-real Element of REAL
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
||.(0. T).|| is V11() real ext-real Element of REAL
N . n is Element of the carrier of S
y1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
L . y1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
(f /* (N * L)) . y1 is Element of the carrier of T
((f /* (N * L)) . y1) - (f /. x0) is Element of the carrier of T
- (f /. x0) is Element of the carrier of T
K192(T,((f /* (N * L)) . y1),(- (f /. x0))) is Element of the carrier of T
||.(((f /* (N * L)) . y1) - (f /. x0)).|| is V11() real ext-real Element of REAL
(N * L) . y1 is Element of the carrier of S
f /. ((N * L) . y1) is Element of the carrier of T
(f /. ((N * L) . y1)) - (f /. x0) is Element of the carrier of T
K192(T,(f /. ((N * L) . y1)),(- (f /. x0))) is Element of the carrier of T
||.((f /. ((N * L) . y1)) - (f /. x0)).|| is V11() real ext-real Element of REAL
f /. (N . n) is Element of the carrier of T
(f /. (N . n)) - (f /. x0) is Element of the carrier of T
K192(T,(f /. (N . n)),(- (f /. x0))) is Element of the carrier of T
||.((f /. (N . n)) - (f /. x0)).|| is V11() real ext-real Element of REAL
(f /* N) . n is Element of the carrier of T
((f /* N) . n) - (f /. x0) is Element of the carrier of T
K192(T,((f /* N) . n),(- (f /. x0))) is Element of the carrier of T
||.(((f /* N) . n) - (f /. x0)).|| is V11() real ext-real Element of REAL
N . n is Element of the carrier of S
(f /* N) . n is Element of the carrier of T
((f /* N) . n) - (f /. x0) is Element of the carrier of T
- (f /. x0) is Element of the carrier of T
K192(T,((f /* N) . n),(- (f /. x0))) is Element of the carrier of T
||.(((f /* N) . n) - (f /. x0)).|| is V11() real ext-real Element of REAL
(f /* N) . n is Element of the carrier of T
((f /* N) . n) - (f /. x0) is Element of the carrier of T
- (f /. x0) is Element of the carrier of T
K192(T,((f /* N) . n),(- (f /. x0))) is Element of the carrier of T
||.(((f /* N) . n) - (f /. x0)).|| is V11() real ext-real Element of REAL
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
K7( the carrier of S, the carrier of T) is set
K6(K7( the carrier of S, the carrier of T)) is set
f is V16() V19( the carrier of S) V20( the carrier of T) Function-like (S,T) Element of K6(K7( the carrier of S, the carrier of T))
x0 is V16() V19( the carrier of S) V20( the carrier of T) Function-like (S,T) Element of K6(K7( the carrier of S, the carrier of T))
f + x0 is V16() V19( the carrier of S) V20( the carrier of T) Function-like Element of K6(K7( the carrier of S, the carrier of T))
f - x0 is V16() V19( the carrier of S) V20( the carrier of T) Function-like Element of K6(K7( the carrier of S, the carrier of T))
0. S is V68(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
K7(NAT, the carrier of S) is set
K6(K7(NAT, the carrier of S)) is set
N is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total non-zero (S, 0. S) Element of K6(K7(NAT, the carrier of S))
f /* N is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
||.N.|| is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
||.N.|| " is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(T,(f /* N),(||.N.|| ")) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
x0 /* N is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(T,(x0 /* N),(||.N.|| ")) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(f + x0) /* N is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(T,((f + x0) /* N),(||.N.|| ")) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(f /* N) + (x0 /* N) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(T,((f /* N) + (x0 /* N)),(||.N.|| ")) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(T,(f /* N),(||.N.|| ")) + (T,(x0 /* N),(||.N.|| ")) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
lim (T,(f /* N),(||.N.|| ")) is Element of the carrier of T
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
lim (T,(x0 /* N),(||.N.|| ")) is Element of the carrier of T
lim (T,((f + x0) /* N),(||.N.|| ")) is Element of the carrier of T
(0. T) + (0. T) is Element of the carrier of T
N is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total non-zero (S, 0. S) Element of K6(K7(NAT, the carrier of S))
f /* N is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
||.N.|| is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
||.N.|| " is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(T,(f /* N),(||.N.|| ")) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
x0 /* N is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(T,(x0 /* N),(||.N.|| ")) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(f - x0) /* N is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(T,((f - x0) /* N),(||.N.|| ")) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(f /* N) - (x0 /* N) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(T,((f /* N) - (x0 /* N)),(||.N.|| ")) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(T,(f /* N),(||.N.|| ")) - (T,(x0 /* N),(||.N.|| ")) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
lim (T,(f /* N),(||.N.|| ")) is Element of the carrier of T
lim (T,(x0 /* N),(||.N.|| ")) is Element of the carrier of T
lim (T,((f - x0) /* N),(||.N.|| ")) is Element of the carrier of T
(0. T) - (0. T) is Element of the carrier of T
- (0. T) is Element of the carrier of T
K192(T,(0. T),(- (0. T))) is Element of the carrier of T
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
K7( the carrier of S, the carrier of T) is set
K6(K7( the carrier of S, the carrier of T)) is set
f is V11() real ext-real Element of REAL
x0 is V16() V19( the carrier of S) V20( the carrier of T) Function-like (S,T) Element of K6(K7( the carrier of S, the carrier of T))
f (#) x0 is V16() V19( the carrier of S) V20( the carrier of T) Function-like Element of K6(K7( the carrier of S, the carrier of T))
0. S is V68(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
K7(NAT, the carrier of S) is set
K6(K7(NAT, the carrier of S)) is set
N is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total non-zero (S, 0. S) Element of K6(K7(NAT, the carrier of S))
x0 /* N is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
||.N.|| is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
||.N.|| " is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(T,(x0 /* N),(||.N.|| ")) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(f (#) x0) /* N is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(T,((f (#) x0) /* N),(||.N.|| ")) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
f * (x0 /* N) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(T,(f * (x0 /* N)),(||.N.|| ")) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
f * (T,(x0 /* N),(||.N.|| ")) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
lim (T,(x0 /* N),(||.N.|| ")) is Element of the carrier of T
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
lim (T,((f (#) x0) /* N),(||.N.|| ")) is Element of the carrier of T
f * (0. T) is Element of the carrier of T
the Mult of T is V16() V19(K7(REAL, the carrier of T)) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of T), the carrier of T))
K7(REAL, the carrier of T) is set
K7(K7(REAL, the carrier of T), the carrier of T) is set
K6(K7(K7(REAL, the carrier of T), the carrier of T)) is set
K249( the Mult of T,f,(0. T)) is set
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
K7( the carrier of T, the carrier of S) is set
K6(K7( the carrier of T, the carrier of S)) is set
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
K7( the carrier of T, the carrier of S) is set
K6(K7( the carrier of T, the carrier of S)) is set
f is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
x0 is Element of the carrier of T
R_NormSpace_of_BoundedLinearOperators (T,S) is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S)) is V1() set
dom f is Element of K6( the carrier of T)
K6( the carrier of T) is set
f /. x0 is Element of the carrier of S
N is Neighbourhood of x0
L is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
R is V16() V19( the carrier of T) V20( the carrier of S) Function-like (T,S) Element of K6(K7( the carrier of T, the carrier of S))
L is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
R is V16() V19( the carrier of T) V20( the carrier of S) Function-like (T,S) Element of K6(K7( the carrier of T, the carrier of S))
N is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
L is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
BoundedLinearOperators (T,S) is V1() Element of K6( the carrier of (R_VectorSpace_of_LinearOperators (T,S)))
R_VectorSpace_of_LinearOperators (T,S) is non empty V87() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators (T,S)) is V1() set
K6( the carrier of (R_VectorSpace_of_LinearOperators (T,S))) is set
Zero_ ((BoundedLinearOperators (T,S)),(R_VectorSpace_of_LinearOperators (T,S))) is Element of BoundedLinearOperators (T,S)
Add_ ((BoundedLinearOperators (T,S)),(R_VectorSpace_of_LinearOperators (T,S))) is V16() V19(K7((BoundedLinearOperators (T,S)),(BoundedLinearOperators (T,S)))) V20( BoundedLinearOperators (T,S)) Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators (T,S)),(BoundedLinearOperators (T,S))),(BoundedLinearOperators (T,S))))
K7((BoundedLinearOperators (T,S)),(BoundedLinearOperators (T,S))) is set
K7(K7((BoundedLinearOperators (T,S)),(BoundedLinearOperators (T,S))),(BoundedLinearOperators (T,S))) is set
K6(K7(K7((BoundedLinearOperators (T,S)),(BoundedLinearOperators (T,S))),(BoundedLinearOperators (T,S)))) is set
Mult_ ((BoundedLinearOperators (T,S)),(R_VectorSpace_of_LinearOperators (T,S))) is V16() V19(K7(REAL,(BoundedLinearOperators (T,S)))) V20( BoundedLinearOperators (T,S)) Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators (T,S))),(BoundedLinearOperators (T,S))))
K7(REAL,(BoundedLinearOperators (T,S))) is set
K7(K7(REAL,(BoundedLinearOperators (T,S))),(BoundedLinearOperators (T,S))) is set
K6(K7(K7(REAL,(BoundedLinearOperators (T,S))),(BoundedLinearOperators (T,S)))) is set
BoundedLinearOperatorsNorm (T,S) is V1() V16() V19( BoundedLinearOperators (T,S)) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators (T,S)),REAL))
K7((BoundedLinearOperators (T,S)),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((BoundedLinearOperators (T,S)),REAL)) is set
NORMSTR(# (BoundedLinearOperators (T,S)),(Zero_ ((BoundedLinearOperators (T,S)),(R_VectorSpace_of_LinearOperators (T,S)))),(Add_ ((BoundedLinearOperators (T,S)),(R_VectorSpace_of_LinearOperators (T,S)))),(Mult_ ((BoundedLinearOperators (T,S)),(R_VectorSpace_of_LinearOperators (T,S)))),(BoundedLinearOperatorsNorm (T,S)) #) is strict NORMSTR
N2 is Neighbourhood of x0
L is V16() V19( the carrier of T) V20( the carrier of S) Function-like (T,S) Element of K6(K7( the carrier of T, the carrier of S))
N2 is Neighbourhood of x0
L is V16() V19( the carrier of T) V20( the carrier of S) Function-like (T,S) Element of K6(K7( the carrier of T, the carrier of S))
L is V16() V19( the carrier of T) V20( the carrier of S) Function-like (T,S) Element of K6(K7( the carrier of T, the carrier of S))
R is Neighbourhood of x0
x is V16() V19( the carrier of T) V20( the carrier of S) Function-like (T,S) Element of K6(K7( the carrier of T, the carrier of S))
R is Neighbourhood of x0
x is V16() V19( the carrier of T) V20( the carrier of S) Function-like (T,S) Element of K6(K7( the carrier of T, the carrier of S))
x is V16() V19( the carrier of T) V20( the carrier of S) Function-like (T,S) Element of K6(K7( the carrier of T, the carrier of S))
R is Element of BoundedLinearOperators (T,S)
modetrans (R,T,S) is V1() V16() V19( the carrier of T) V20( the carrier of S) Function-like total quasi_total V155(T,S) V156(T,S) Lipschitzian Element of K6(K7( the carrier of T, the carrier of S))
s1 is Element of BoundedLinearOperators (T,S)
modetrans (s1,T,S) is V1() V16() V19( the carrier of T) V20( the carrier of S) Function-like total quasi_total V155(T,S) V156(T,S) Lipschitzian Element of K6(K7( the carrier of T, the carrier of S))
y is Element of the carrier of T
(modetrans (R,T,S)) . y is Element of the carrier of S
(modetrans (s1,T,S)) . y is Element of the carrier of S
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
0 * y is Element of the carrier of T
the Mult of T is V16() V19(K7(REAL, the carrier of T)) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of T), the carrier of T))
K7(REAL, the carrier of T) is set
K7(K7(REAL, the carrier of T), the carrier of T) is set
K6(K7(K7(REAL, the carrier of T), the carrier of T)) is set
K249( the Mult of T,0,y) is set
(modetrans (R,T,S)) . (0 * y) is Element of the carrier of S
0 * ((modetrans (R,T,S)) . y) is Element of the carrier of S
the Mult of S is V16() V19(K7(REAL, the carrier of S)) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of S), the carrier of S))
K7(REAL, the carrier of S) is set
K7(K7(REAL, the carrier of S), the carrier of S) is set
K6(K7(K7(REAL, the carrier of S), the carrier of S)) is set
K249( the Mult of S,0,((modetrans (R,T,S)) . y)) is set
0. S is V68(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
0 * ((modetrans (s1,T,S)) . y) is Element of the carrier of S
K249( the Mult of S,0,((modetrans (s1,T,S)) . y)) is set
(modetrans (s1,T,S)) . (0 * y) is Element of the carrier of S
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
n is Neighbourhood of x0
y1 is V11() real ext-real Element of REAL
{ b₁ where b₁ is Element of the carrier of T : not y1 <= ||.(b₁ - x0).|| } is set
||.y.|| is V11() real ext-real Element of REAL
||.y.|| / y1 is V11() real ext-real Element of REAL
y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
y + 1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
(y + 1) * y1 is V11() real ext-real Element of REAL
(||.y.|| / y1) * y1 is V11() real ext-real Element of REAL
||.y.|| / (y + 1) is V11() real ext-real Element of REAL
x1 is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
lim x1 is V11() real ext-real Element of REAL
x2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
x1 . x2 is V11() real ext-real Element of REAL
x2 + (y + 1) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
1 / (x2 + (y + 1)) is V11() real ext-real Element of REAL
(x1 . x2) * y is Element of the carrier of T
the Mult of T is V16() V19(K7(REAL, the carrier of T)) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of T), the carrier of T))
K7(REAL, the carrier of T) is set
K7(K7(REAL, the carrier of T), the carrier of T) is set
K6(K7(K7(REAL, the carrier of T), the carrier of T)) is set
K249( the Mult of T,(x1 . x2),y) is set
((x1 . x2) * y) + x0 is Element of the carrier of T
||.y.|| / (x2 + (y + 1)) is V11() real ext-real Element of REAL
(x2 + (y + 1)) " is V11() real ext-real Element of REAL
1 * ((x2 + (y + 1)) ") is V11() real ext-real Element of REAL
(((x1 . x2) * y) + x0) - x0 is Element of the carrier of T
- x0 is Element of the carrier of T
K192(T,(((x1 . x2) * y) + x0),(- x0)) is Element of the carrier of T
||.((((x1 . x2) * y) + x0) - x0).|| is V11() real ext-real Element of REAL
x0 - x0 is Element of the carrier of T
K192(T,x0,(- x0)) is Element of the carrier of T
((x1 . x2) * y) + (x0 - x0) is Element of the carrier of T
||.(((x1 . x2) * y) + (x0 - x0)).|| is V11() real ext-real Element of REAL
((x1 . x2) * y) + (0. T) is Element of the carrier of T
||.(((x1 . x2) * y) + (0. T)).|| is V11() real ext-real Element of REAL
||.((x1 . x2) * y).|| is V11() real ext-real Element of REAL
(1 / (x2 + (y + 1))) * y is Element of the carrier of T
K249( the Mult of T,(1 / (x2 + (y + 1))),y) is set
||.((1 / (x2 + (y + 1))) * y).|| is V11() real ext-real Element of REAL
abs (1 / (x2 + (y + 1))) is V11() real ext-real Element of REAL
(abs (1 / (x2 + (y + 1)))) * ||.y.|| is V11() real ext-real Element of REAL
(1 / (x2 + (y + 1))) * ||.y.|| is V11() real ext-real Element of REAL
x1 is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
lim x1 is V11() real ext-real Element of REAL
x1 is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
lim x1 is V11() real ext-real Element of REAL
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
x2 is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
s is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
x2 . s is Element of the carrier of T
x1 . s is V11() real ext-real Element of REAL
(x1 . s) * y is Element of the carrier of T
the Mult of T is V16() V19(K7(REAL, the carrier of T)) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of T), the carrier of T))
K7(REAL, the carrier of T) is set
K7(K7(REAL, the carrier of T), the carrier of T) is set
K6(K7(K7(REAL, the carrier of T), the carrier of T)) is set
K249( the Mult of T,(x1 . s),y) is set
s is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
x2 . s is Element of the carrier of T
x1 . s is V11() real ext-real Element of REAL
(x1 . s) * y is Element of the carrier of T
K249( the Mult of T,(x1 . s),y) is set
s + (y + 1) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
1 / (s + (y + 1)) is V11() real ext-real Element of REAL
(1 / (s + (y + 1))) * y is Element of the carrier of T
K249( the Mult of T,(1 / (s + (y + 1))),y) is set
lim x2 is Element of the carrier of T
n is Element of the carrier of T
f /. n is Element of the carrier of S
(f /. n) - (f /. x0) is Element of the carrier of S
- (f /. x0) is Element of the carrier of S
K192(S,(f /. n),(- (f /. x0))) is Element of the carrier of S
n - x0 is Element of the carrier of T
- x0 is Element of the carrier of T
K192(T,n,(- x0)) is Element of the carrier of T
R . (n - x0) is Element of the carrier of S
L /. (n - x0) is Element of the carrier of S
(R . (n - x0)) + (L /. (n - x0)) is Element of the carrier of S
s1 . (n - x0) is Element of the carrier of S
x /. (n - x0) is Element of the carrier of S
(s1 . (n - x0)) + (x /. (n - x0)) is Element of the carrier of S
dom x is Element of K6( the carrier of T)
s is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total non-zero (T, 0. T) Element of K6(K7(NAT, the carrier of T))
rng s is Element of K6( the carrier of T)
dom L is Element of K6( the carrier of T)
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
x1 . n is V11() real ext-real Element of REAL
(x1 . n) * y is Element of the carrier of T
K249( the Mult of T,(x1 . n),y) is set
((x1 . n) * y) + x0 is Element of the carrier of T
(((x1 . n) * y) + x0) - x0 is Element of the carrier of T
K192(T,(((x1 . n) * y) + x0),(- x0)) is Element of the carrier of T
x0 - x0 is Element of the carrier of T
K192(T,x0,(- x0)) is Element of the carrier of T
((x1 . n) * y) + (x0 - x0) is Element of the carrier of T
((x1 . n) * y) + (0. T) is Element of the carrier of T
s . n is Element of the carrier of T
||.(s . n).|| is V11() real ext-real Element of REAL
||.((x1 . n) * y).|| is V11() real ext-real Element of REAL
abs (x1 . n) is V11() real ext-real Element of REAL
(abs (x1 . n)) * ||.y.|| is V11() real ext-real Element of REAL
(x1 . n) * ||.y.|| is V11() real ext-real Element of REAL
(x1 . n) " is V11() real ext-real Element of REAL
((x1 . n) ") * ||.(s . n).|| is V11() real ext-real Element of REAL
((x1 . n) ") * (x1 . n) is V11() real ext-real Element of REAL
(((x1 . n) ") * (x1 . n)) * ||.y.|| is V11() real ext-real Element of REAL
(x1 . n) / (x1 . n) is V11() real ext-real Element of REAL
((x1 . n) / (x1 . n)) * ||.y.|| is V11() real ext-real Element of REAL
1 * ||.y.|| is V11() real ext-real Element of REAL
||.(s . n).|| " is V11() real ext-real Element of REAL
||.y.|| * (||.(s . n).|| ") is V11() real ext-real Element of REAL
||.(s . n).|| * (||.(s . n).|| ") is V11() real ext-real Element of REAL
((x1 . n) ") * (||.(s . n).|| * (||.(s . n).|| ")) is V11() real ext-real Element of REAL
||.(s . n).|| / ||.(s . n).|| is V11() real ext-real Element of REAL
((x1 . n) ") * (||.(s . n).|| / ||.(s . n).||) is V11() real ext-real Element of REAL
((x1 . n) ") * 1 is V11() real ext-real Element of REAL
||.s.|| is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
||.s.|| . n is V11() real ext-real Element of REAL
(||.s.|| . n) " is V11() real ext-real Element of REAL
||.y.|| * ((||.s.|| . n) ") is V11() real ext-real Element of REAL
R . ((x1 . n) * y) is Element of the carrier of S
L /. ((x1 . n) * y) is Element of the carrier of S
(R . ((x1 . n) * y)) + (L /. ((x1 . n) * y)) is Element of the carrier of S
s1 . ((x1 . n) * y) is Element of the carrier of S
x /. ((x1 . n) * y) is Element of the carrier of S
(s1 . ((x1 . n) * y)) + (x /. ((x1 . n) * y)) is Element of the carrier of S
((x1 . n) ") * (R . ((x1 . n) * y)) is Element of the carrier of S
the Mult of S is V16() V19(K7(REAL, the carrier of S)) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of S), the carrier of S))
K7(REAL, the carrier of S) is set
K7(K7(REAL, the carrier of S), the carrier of S) is set
K6(K7(K7(REAL, the carrier of S), the carrier of S)) is set
K249( the Mult of S,((x1 . n) "),(R . ((x1 . n) * y))) is set
((x1 . n) ") * (L /. ((x1 . n) * y)) is Element of the carrier of S
K249( the Mult of S,((x1 . n) "),(L /. ((x1 . n) * y))) is set
(((x1 . n) ") * (R . ((x1 . n) * y))) + (((x1 . n) ") * (L /. ((x1 . n) * y))) is Element of the carrier of S
((x1 . n) ") * ((s1 . ((x1 . n) * y)) + (x /. ((x1 . n) * y))) is Element of the carrier of S
K249( the Mult of S,((x1 . n) "),((s1 . ((x1 . n) * y)) + (x /. ((x1 . n) * y)))) is set
((x1 . n) ") * (s1 . ((x1 . n) * y)) is Element of the carrier of S
K249( the Mult of S,((x1 . n) "),(s1 . ((x1 . n) * y))) is set
(modetrans (s1,T,S)) . ((x1 . n) * y) is Element of the carrier of S
((x1 . n) ") * ((modetrans (s1,T,S)) . ((x1 . n) * y)) is Element of the carrier of S
K249( the Mult of S,((x1 . n) "),((modetrans (s1,T,S)) . ((x1 . n) * y))) is set
(x1 . n) * ((modetrans (s1,T,S)) . y) is Element of the carrier of S
K249( the Mult of S,(x1 . n),((modetrans (s1,T,S)) . y)) is set
((x1 . n) ") * ((x1 . n) * ((modetrans (s1,T,S)) . y)) is Element of the carrier of S
K249( the Mult of S,((x1 . n) "),((x1 . n) * ((modetrans (s1,T,S)) . y))) is set
s1 . y is Element of the carrier of S
(x1 . n) * (s1 . y) is Element of the carrier of S
K249( the Mult of S,(x1 . n),(s1 . y)) is set
((x1 . n) ") * ((x1 . n) * (s1 . y)) is Element of the carrier of S
K249( the Mult of S,((x1 . n) "),((x1 . n) * (s1 . y))) is set
(((x1 . n) ") * (x1 . n)) * (s1 . y) is Element of the carrier of S
K249( the Mult of S,(((x1 . n) ") * (x1 . n)),(s1 . y)) is set
((x1 . n) / (x1 . n)) * (s1 . y) is Element of the carrier of S
K249( the Mult of S,((x1 . n) / (x1 . n)),(s1 . y)) is set
1 * (s1 . y) is Element of the carrier of S
K249( the Mult of S,1,(s1 . y)) is set
(modetrans (R,T,S)) . ((x1 . n) * y) is Element of the carrier of S
((x1 . n) ") * ((modetrans (R,T,S)) . ((x1 . n) * y)) is Element of the carrier of S
K249( the Mult of S,((x1 . n) "),((modetrans (R,T,S)) . ((x1 . n) * y))) is set
(x1 . n) * ((modetrans (R,T,S)) . y) is Element of the carrier of S
K249( the Mult of S,(x1 . n),((modetrans (R,T,S)) . y)) is set
((x1 . n) ") * ((x1 . n) * ((modetrans (R,T,S)) . y)) is Element of the carrier of S
K249( the Mult of S,((x1 . n) "),((x1 . n) * ((modetrans (R,T,S)) . y))) is set
R . y is Element of the carrier of S
(x1 . n) * (R . y) is Element of the carrier of S
K249( the Mult of S,(x1 . n),(R . y)) is set
((x1 . n) ") * ((x1 . n) * (R . y)) is Element of the carrier of S
K249( the Mult of S,((x1 . n) "),((x1 . n) * (R . y))) is set
(((x1 . n) ") * (x1 . n)) * (R . y) is Element of the carrier of S
K249( the Mult of S,(((x1 . n) ") * (x1 . n)),(R . y)) is set
((x1 . n) / (x1 . n)) * (R . y) is Element of the carrier of S
K249( the Mult of S,((x1 . n) / (x1 . n)),(R . y)) is set
1 * (R . y) is Element of the carrier of S
K249( the Mult of S,1,(R . y)) is set
(R . y) + (((x1 . n) ") * (L /. ((x1 . n) * y))) is Element of the carrier of S
((x1 . n) ") * (x /. ((x1 . n) * y)) is Element of the carrier of S
K249( the Mult of S,((x1 . n) "),(x /. ((x1 . n) * y))) is set
(s1 . y) + (((x1 . n) ") * (x /. ((x1 . n) * y))) is Element of the carrier of S
x /. (s . n) is Element of the carrier of S
((x1 . n) ") * (x /. (s . n)) is Element of the carrier of S
K249( the Mult of S,((x1 . n) "),(x /. (s . n))) is set
||.s.|| " is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(||.s.|| ") . n is V11() real ext-real Element of REAL
||.y.|| * ((||.s.|| ") . n) is V11() real ext-real Element of REAL
(||.y.|| * ((||.s.|| ") . n)) * (x /. (s . n)) is Element of the carrier of S
K249( the Mult of S,(||.y.|| * ((||.s.|| ") . n)),(x /. (s . n))) is set
((||.s.|| ") . n) * (x /. (s . n)) is Element of the carrier of S
K249( the Mult of S,((||.s.|| ") . n),(x /. (s . n))) is set
||.y.|| * (((||.s.|| ") . n) * (x /. (s . n))) is Element of the carrier of S
K249( the Mult of S,||.y.||,(((||.s.|| ") . n) * (x /. (s . n)))) is set
x /* s is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
K7(NAT, the carrier of S) is set
K6(K7(NAT, the carrier of S)) is set
(x /* s) . n is Element of the carrier of S
((||.s.|| ") . n) * ((x /* s) . n) is Element of the carrier of S
K249( the Mult of S,((||.s.|| ") . n),((x /* s) . n)) is set
||.y.|| * (((||.s.|| ") . n) * ((x /* s) . n)) is Element of the carrier of S
K249( the Mult of S,||.y.||,(((||.s.|| ") . n) * ((x /* s) . n))) is set
(S,(x /* s),(||.s.|| ")) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
(S,(x /* s),(||.s.|| ")) . n is Element of the carrier of S
||.y.|| * ((S,(x /* s),(||.s.|| ")) . n) is Element of the carrier of S
K249( the Mult of S,||.y.||,((S,(x /* s),(||.s.|| ")) . n)) is set
L /. (s . n) is Element of the carrier of S
((x1 . n) ") * (L /. (s . n)) is Element of the carrier of S
K249( the Mult of S,((x1 . n) "),(L /. (s . n))) is set
(||.y.|| * ((||.s.|| ") . n)) * (L /. (s . n)) is Element of the carrier of S
K249( the Mult of S,(||.y.|| * ((||.s.|| ") . n)),(L /. (s . n))) is set
((||.s.|| ") . n) * (L /. (s . n)) is Element of the carrier of S
K249( the Mult of S,((||.s.|| ") . n),(L /. (s . n))) is set
||.y.|| * (((||.s.|| ") . n) * (L /. (s . n))) is Element of the carrier of S
K249( the Mult of S,||.y.||,(((||.s.|| ") . n) * (L /. (s . n)))) is set
L /* s is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
(L /* s) . n is Element of the carrier of S
((||.s.|| ") . n) * ((L /* s) . n) is Element of the carrier of S
K249( the Mult of S,((||.s.|| ") . n),((L /* s) . n)) is set
||.y.|| * (((||.s.|| ") . n) * ((L /* s) . n)) is Element of the carrier of S
K249( the Mult of S,||.y.||,(((||.s.|| ") . n) * ((L /* s) . n))) is set
(S,(L /* s),(||.s.|| ")) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
(S,(L /* s),(||.s.|| ")) . n is Element of the carrier of S
||.y.|| * ((S,(L /* s),(||.s.|| ")) . n) is Element of the carrier of S
K249( the Mult of S,||.y.||,((S,(L /* s),(||.s.|| ")) . n)) is set
(||.y.|| * ((S,(L /* s),(||.s.|| ")) . n)) - (||.y.|| * ((S,(L /* s),(||.s.|| ")) . n)) is Element of the carrier of S
- (||.y.|| * ((S,(L /* s),(||.s.|| ")) . n)) is Element of the carrier of S
K192(S,(||.y.|| * ((S,(L /* s),(||.s.|| ")) . n)),(- (||.y.|| * ((S,(L /* s),(||.s.|| ")) . n)))) is Element of the carrier of S
(R . y) + ((||.y.|| * ((S,(L /* s),(||.s.|| ")) . n)) - (||.y.|| * ((S,(L /* s),(||.s.|| ")) . n))) is Element of the carrier of S
(s1 . y) + (||.y.|| * ((S,(x /* s),(||.s.|| ")) . n)) is Element of the carrier of S
((s1 . y) + (||.y.|| * ((S,(x /* s),(||.s.|| ")) . n))) - (||.y.|| * ((S,(L /* s),(||.s.|| ")) . n)) is Element of the carrier of S
K192(S,((s1 . y) + (||.y.|| * ((S,(x /* s),(||.s.|| ")) . n))),(- (||.y.|| * ((S,(L /* s),(||.s.|| ")) . n)))) is Element of the carrier of S
0. S is V68(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
(R . y) + (0. S) is Element of the carrier of S
(R . y) - (s1 . y) is Element of the carrier of S
- (s1 . y) is Element of the carrier of S
K192(S,(R . y),(- (s1 . y))) is Element of the carrier of S
(||.y.|| * ((S,(x /* s),(||.s.|| ")) . n)) - (||.y.|| * ((S,(L /* s),(||.s.|| ")) . n)) is Element of the carrier of S
K192(S,(||.y.|| * ((S,(x /* s),(||.s.|| ")) . n)),(- (||.y.|| * ((S,(L /* s),(||.s.|| ")) . n)))) is Element of the carrier of S
(s1 . y) + ((||.y.|| * ((S,(x /* s),(||.s.|| ")) . n)) - (||.y.|| * ((S,(L /* s),(||.s.|| ")) . n))) is Element of the carrier of S
(- (s1 . y)) + ((s1 . y) + ((||.y.|| * ((S,(x /* s),(||.s.|| ")) . n)) - (||.y.|| * ((S,(L /* s),(||.s.|| ")) . n)))) is Element of the carrier of S
(- (s1 . y)) + (s1 . y) is Element of the carrier of S
((- (s1 . y)) + (s1 . y)) + ((||.y.|| * ((S,(x /* s),(||.s.|| ")) . n)) - (||.y.|| * ((S,(L /* s),(||.s.|| ")) . n))) is Element of the carrier of S
(0. S) + ((||.y.|| * ((S,(x /* s),(||.s.|| ")) . n)) - (||.y.|| * ((S,(L /* s),(||.s.|| ")) . n))) is Element of the carrier of S
((S,(x /* s),(||.s.|| ")) . n) - ((S,(L /* s),(||.s.|| ")) . n) is Element of the carrier of S
- ((S,(L /* s),(||.s.|| ")) . n) is Element of the carrier of S
K192(S,((S,(x /* s),(||.s.|| ")) . n),(- ((S,(L /* s),(||.s.|| ")) . n))) is Element of the carrier of S
||.y.|| * (((S,(x /* s),(||.s.|| ")) . n) - ((S,(L /* s),(||.s.|| ")) . n)) is Element of the carrier of S
K249( the Mult of S,||.y.||,(((S,(x /* s),(||.s.|| ")) . n) - ((S,(L /* s),(||.s.|| ")) . n))) is set
(S,(x /* s),(||.s.|| ")) - (S,(L /* s),(||.s.|| ")) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
((S,(x /* s),(||.s.|| ")) - (S,(L /* s),(||.s.|| "))) . n is Element of the carrier of S
||.y.|| * (((S,(x /* s),(||.s.|| ")) - (S,(L /* s),(||.s.|| "))) . n) is Element of the carrier of S
K249( the Mult of S,||.y.||,(((S,(x /* s),(||.s.|| ")) - (S,(L /* s),(||.s.|| "))) . n)) is set
||.y.|| * ((S,(x /* s),(||.s.|| ")) - (S,(L /* s),(||.s.|| "))) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
(||.y.|| * ((S,(x /* s),(||.s.|| ")) - (S,(L /* s),(||.s.|| ")))) . n is Element of the carrier of S
(||.y.|| * ((S,(x /* s),(||.s.|| ")) - (S,(L /* s),(||.s.|| ")))) . 1 is Element of the carrier of S
lim (||.y.|| * ((S,(x /* s),(||.s.|| ")) - (S,(L /* s),(||.s.|| ")))) is Element of the carrier of S
lim (S,(L /* s),(||.s.|| ")) is Element of the carrier of S
lim (S,(x /* s),(||.s.|| ")) is Element of the carrier of S
lim ((S,(x /* s),(||.s.|| ")) - (S,(L /* s),(||.s.|| "))) is Element of the carrier of S
(0. S) - (0. S) is Element of the carrier of S
- (0. S) is Element of the carrier of S
K192(S,(0. S),(- (0. S))) is Element of the carrier of S
||.y.|| * (lim ((S,(x /* s),(||.s.|| ")) - (S,(L /* s),(||.s.|| ")))) is Element of the carrier of S
K249( the Mult of S,||.y.||,(lim ((S,(x /* s),(||.s.|| ")) - (S,(L /* s),(||.s.|| "))))) is set
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
f is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of f is V1() non trivial set
K7( the carrier of S, the carrier of f) is set
K6(K7( the carrier of S, the carrier of f)) is set
T is set
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
K6( the carrier of S) is set
f is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of f is V1() non trivial set
K7( the carrier of S, the carrier of f) is set
K6(K7( the carrier of S, the carrier of f)) is set
x0 is V16() V19( the carrier of S) V20( the carrier of f) Function-like Element of K6(K7( the carrier of S, the carrier of f))
dom x0 is Element of K6( the carrier of S)
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
K6( the carrier of T) is set
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
K7( the carrier of T, the carrier of S) is set
K6(K7( the carrier of T, the carrier of S)) is set
f is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
dom f is Element of K6( the carrier of T)
x0 is Element of K6( the carrier of T)
N is Element of the carrier of T
f | x0 is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
dom (f | x0) is Element of K6( the carrier of T)
R_NormSpace_of_BoundedLinearOperators (T,S) is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S)) is V1() set
(f | x0) /. N is Element of the carrier of S
L is Neighbourhood of N
R is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
s1 is V16() V19( the carrier of T) V20( the carrier of S) Function-like (T,S) Element of K6(K7( the carrier of T, the carrier of S))
R is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
s1 is V16() V19( the carrier of T) V20( the carrier of S) Function-like (T,S) Element of K6(K7( the carrier of T, the carrier of S))
f /. N is Element of the carrier of S
(dom f) /\ x0 is Element of K6( the carrier of T)
N2 is Element of the carrier of T
f /. N2 is Element of the carrier of S
(f /. N2) - (f /. N) is Element of the carrier of S
- (f /. N) is Element of the carrier of S
K192(S,(f /. N2),(- (f /. N))) is Element of the carrier of S
N2 - N is Element of the carrier of T
- N is Element of the carrier of T
K192(T,N2,(- N)) is Element of the carrier of T
R . (N2 - N) is Element of the carrier of S
s1 /. (N2 - N) is Element of the carrier of S
(R . (N2 - N)) + (s1 /. (N2 - N)) is Element of the carrier of S
(f | x0) /. N2 is Element of the carrier of S
((f | x0) /. N2) - ((f | x0) /. N) is Element of the carrier of S
- ((f | x0) /. N) is Element of the carrier of S
K192(S,((f | x0) /. N2),(- ((f | x0) /. N))) is Element of the carrier of S
(f /. N2) - ((f | x0) /. N) is Element of the carrier of S
K192(S,(f /. N2),(- ((f | x0) /. N))) is Element of the carrier of S
f | x0 is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
N is Element of the carrier of T
L is Neighbourhood of N
R_NormSpace_of_BoundedLinearOperators (T,S) is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S)) is V1() set
f /. N is Element of the carrier of S
R is Neighbourhood of N
s1 is Neighbourhood of N
dom (f | x0) is Element of K6( the carrier of T)
(f | x0) /. N is Element of the carrier of S
(dom f) /\ x0 is Element of K6( the carrier of T)
N2 is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
L is V16() V19( the carrier of T) V20( the carrier of S) Function-like (T,S) Element of K6(K7( the carrier of T, the carrier of S))
N2 is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
L is V16() V19( the carrier of T) V20( the carrier of S) Function-like (T,S) Element of K6(K7( the carrier of T, the carrier of S))
R is Element of the carrier of T
(f | x0) /. R is Element of the carrier of S
((f | x0) /. R) - ((f | x0) /. N) is Element of the carrier of S
- ((f | x0) /. N) is Element of the carrier of S
K192(S,((f | x0) /. R),(- ((f | x0) /. N))) is Element of the carrier of S
R - N is Element of the carrier of T
- N is Element of the carrier of T
K192(T,R,(- N)) is Element of the carrier of T
N2 . (R - N) is Element of the carrier of S
L /. (R - N) is Element of the carrier of S
(N2 . (R - N)) + (L /. (R - N)) is Element of the carrier of S
f /. R is Element of the carrier of S
(f /. R) - (f /. N) is Element of the carrier of S
- (f /. N) is Element of the carrier of S
K192(S,(f /. R),(- (f /. N))) is Element of the carrier of S
((f | x0) /. R) - (f /. N) is Element of the carrier of S
K192(S,((f | x0) /. R),(- (f /. N))) is Element of the carrier of S
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
K6( the carrier of T) is set
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
K7( the carrier of T, the carrier of S) is set
K6(K7( the carrier of T, the carrier of S)) is set
f is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
x0 is Element of K6( the carrier of T)
N is Element of the carrier of T
f | x0 is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
dom (f | x0) is Element of K6( the carrier of T)
R_NormSpace_of_BoundedLinearOperators (T,S) is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S)) is V1() set
(f | x0) /. N is Element of the carrier of S
L is Neighbourhood of N
dom f is Element of K6( the carrier of T)
(dom f) /\ x0 is Element of K6( the carrier of T)
R is Neighbourhood of N
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
K7( the carrier of T, the carrier of S) is set
K6(K7( the carrier of T, the carrier of S)) is set
x0 is set
f is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
R_NormSpace_of_BoundedLinearOperators (T,S) is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S)) is V1() set
K7( the carrier of T, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))) is set
K6(K7( the carrier of T, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S)))) is set
N is V16() V19( the carrier of T) V20( the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))) Function-like Element of K6(K7( the carrier of T, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))))
dom N is Element of K6( the carrier of T)
K6( the carrier of T) is set
L is set
L is set
L is Element of the carrier of T
N . L is set
(T,S,f,L) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
N /. L is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
L is Element of the carrier of T
N /. L is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
(T,S,f,L) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
N is V16() V19( the carrier of T) V20( the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))) Function-like Element of K6(K7( the carrier of T, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))))
dom N is Element of K6( the carrier of T)
K6( the carrier of T) is set
L is V16() V19( the carrier of T) V20( the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))) Function-like Element of K6(K7( the carrier of T, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))))
dom L is Element of K6( the carrier of T)
R is Element of the carrier of T
N /. R is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
(T,S,f,R) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
L /. R is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
K6( the carrier of T) is set
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
K7( the carrier of T, the carrier of S) is set
K6(K7( the carrier of T, the carrier of S)) is set
R_NormSpace_of_BoundedLinearOperators (T,S) is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S)) is V1() set
0. (R_NormSpace_of_BoundedLinearOperators (T,S)) is V16() Function-like V68( R_NormSpace_of_BoundedLinearOperators (T,S)) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
the ZeroF of (R_NormSpace_of_BoundedLinearOperators (T,S)) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
f is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
dom f is Element of K6( the carrier of T)
rng f is Element of K6( the carrier of S)
K6( the carrier of S) is set
x0 is Element of K6( the carrier of T)
(T,S,f,x0) is V16() V19( the carrier of T) V20( the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))) Function-like Element of K6(K7( the carrier of T, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))))
K7( the carrier of T, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))) is set
K6(K7( the carrier of T, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S)))) is set
0. S is V68(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
the carrier of T --> (0. S) is V1() V16() V19( the carrier of T) V20( the carrier of S) Function-like constant total quasi_total Element of K6(K7( the carrier of T, the carrier of S))
R is Element of the carrier of S
{R} is set
BoundedLinearOperators (T,S) is V1() Element of K6( the carrier of (R_VectorSpace_of_LinearOperators (T,S)))
R_VectorSpace_of_LinearOperators (T,S) is non empty V87() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators (T,S)) is V1() set
K6( the carrier of (R_VectorSpace_of_LinearOperators (T,S))) is set
Zero_ ((BoundedLinearOperators (T,S)),(R_VectorSpace_of_LinearOperators (T,S))) is Element of BoundedLinearOperators (T,S)
Add_ ((BoundedLinearOperators (T,S)),(R_VectorSpace_of_LinearOperators (T,S))) is V16() V19(K7((BoundedLinearOperators (T,S)),(BoundedLinearOperators (T,S)))) V20( BoundedLinearOperators (T,S)) Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators (T,S)),(BoundedLinearOperators (T,S))),(BoundedLinearOperators (T,S))))
K7((BoundedLinearOperators (T,S)),(BoundedLinearOperators (T,S))) is set
K7(K7((BoundedLinearOperators (T,S)),(BoundedLinearOperators (T,S))),(BoundedLinearOperators (T,S))) is set
K6(K7(K7((BoundedLinearOperators (T,S)),(BoundedLinearOperators (T,S))),(BoundedLinearOperators (T,S)))) is set
Mult_ ((BoundedLinearOperators (T,S)),(R_VectorSpace_of_LinearOperators (T,S))) is V16() V19(K7(REAL,(BoundedLinearOperators (T,S)))) V20( BoundedLinearOperators (T,S)) Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators (T,S))),(BoundedLinearOperators (T,S))))
K7(REAL,(BoundedLinearOperators (T,S))) is set
K7(K7(REAL,(BoundedLinearOperators (T,S))),(BoundedLinearOperators (T,S))) is set
K6(K7(K7(REAL,(BoundedLinearOperators (T,S))),(BoundedLinearOperators (T,S)))) is set
BoundedLinearOperatorsNorm (T,S) is V1() V16() V19( BoundedLinearOperators (T,S)) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators (T,S)),REAL))
K7((BoundedLinearOperators (T,S)),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((BoundedLinearOperators (T,S)),REAL)) is set
NORMSTR(# (BoundedLinearOperators (T,S)),(Zero_ ((BoundedLinearOperators (T,S)),(R_VectorSpace_of_LinearOperators (T,S)))),(Add_ ((BoundedLinearOperators (T,S)),(R_VectorSpace_of_LinearOperators (T,S)))),(Mult_ ((BoundedLinearOperators (T,S)),(R_VectorSpace_of_LinearOperators (T,S)))),(BoundedLinearOperatorsNorm (T,S)) #) is strict NORMSTR
N is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
dom N is Element of K6( the carrier of T)
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
N2 is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total non-zero (T, 0. T) Element of K6(K7(NAT, the carrier of T))
L is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
N2 . L is Element of the carrier of T
N /. (N2 . L) is Element of the carrier of S
N . (N2 . L) is set
rng N2 is Element of K6( the carrier of T)
N /* N2 is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
K7(NAT, the carrier of S) is set
K6(K7(NAT, the carrier of S)) is set
||.N2.|| is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
||.N2.|| " is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(S,(N /* N2),(||.N2.|| ")) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
(S,(N /* N2),(||.N2.|| ")) . L is Element of the carrier of S
(N /* N2) . L is Element of the carrier of S
(||.N2.|| ") . L is V11() real ext-real Element of REAL
((||.N2.|| ") . L) * ((N /* N2) . L) is Element of the carrier of S
the Mult of S is V16() V19(K7(REAL, the carrier of S)) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of S), the carrier of S))
K7(REAL, the carrier of S) is set
K7(K7(REAL, the carrier of S), the carrier of S) is set
K6(K7(K7(REAL, the carrier of S), the carrier of S)) is set
K249( the Mult of S,((||.N2.|| ") . L),((N /* N2) . L)) is set
((||.N2.|| ") . L) * (N /. (N2 . L)) is Element of the carrier of S
K249( the Mult of S,((||.N2.|| ") . L),(N /. (N2 . L))) is set
(S,(N /* N2),(||.N2.|| ")) . 0 is Element of the carrier of S
lim (S,(N /* N2),(||.N2.|| ")) is Element of the carrier of S
N2 is Element of the carrier of T
f . N2 is set
f /. N2 is Element of the carrier of S
s1 is Element of BoundedLinearOperators (T,S)
L is Element of the carrier of T
R is Neighbourhood of L
f /. L is Element of the carrier of S
N2 is V16() V19( the carrier of T) V20( the carrier of S) Function-like (T,S) Element of K6(K7( the carrier of T, the carrier of S))
x is Element of the carrier of T
f /. x is Element of the carrier of S
(f /. x) - (f /. L) is Element of the carrier of S
- (f /. L) is Element of the carrier of S
K192(S,(f /. x),(- (f /. L))) is Element of the carrier of S
x - L is Element of the carrier of T
- L is Element of the carrier of T
K192(T,x,(- L)) is Element of the carrier of T
s1 . (x - L) is Element of the carrier of S
N2 /. (x - L) is Element of the carrier of S
(s1 . (x - L)) + (N2 /. (x - L)) is Element of the carrier of S
N2 . (x - L) is set
R - (f /. L) is Element of the carrier of S
K192(S,R,(- (f /. L))) is Element of the carrier of S
R - R is Element of the carrier of S
- R is Element of the carrier of S
K192(S,R,(- R)) is Element of the carrier of S
(0. S) + (0. S) is Element of the carrier of S
L is Element of the carrier of T
(T,S,f,x0) /. L is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
f /. L is Element of the carrier of S
R is Neighbourhood of L
x is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
y is V16() V19( the carrier of T) V20( the carrier of S) Function-like (T,S) Element of K6(K7( the carrier of T, the carrier of S))
R is Neighbourhood of L
x is Element of the carrier of T
f /. x is Element of the carrier of S
(f /. x) - (f /. L) is Element of the carrier of S
- (f /. L) is Element of the carrier of S
K192(S,(f /. x),(- (f /. L))) is Element of the carrier of S
x - L is Element of the carrier of T
- L is Element of the carrier of T
K192(T,x,(- L)) is Element of the carrier of T
s1 . (x - L) is Element of the carrier of S
N2 /. (x - L) is Element of the carrier of S
(s1 . (x - L)) + (N2 /. (x - L)) is Element of the carrier of S
N2 . (x - L) is set
R - (f /. L) is Element of the carrier of S
K192(S,R,(- (f /. L))) is Element of the carrier of S
R - R is Element of the carrier of S
- R is Element of the carrier of S
K192(S,R,(- R)) is Element of the carrier of S
(0. S) + (0. S) is Element of the carrier of S
(T,S,f,L) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
S is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total non-zero (T, 0. T) Element of K6(K7(NAT, the carrier of T))
f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
S ^\ f is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total subsequence of S
lim S is Element of the carrier of T
lim (S ^\ f) is Element of the carrier of T
x0 is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
K7( the carrier of T, the carrier of S) is set
K6(K7( the carrier of T, the carrier of S)) is set
0. S is V68(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
f is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
dom f is Element of K6( the carrier of T)
K6( the carrier of T) is set
x0 is Element of the carrier of T
{x0} is set
N is Neighbourhood of x0
L is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total non-zero (T, 0. T) Element of K6(K7(NAT, the carrier of T))
R is V1() V16() V19( NAT ) V20( the carrier of T) Function-like constant total quasi_total Element of K6(K7(NAT, the carrier of T))
rng R is trivial Element of K6( the carrier of T)
L + R is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
rng (L + R) is Element of K6( the carrier of T)
f /* (L + R) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
K7(NAT, the carrier of S) is set
K6(K7(NAT, the carrier of S)) is set
f /* R is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
(f /* (L + R)) - (f /* R) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
lim ((f /* (L + R)) - (f /* R)) is Element of the carrier of S
R_NormSpace_of_BoundedLinearOperators (T,S) is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S)) is V1() set
f /. x0 is Element of the carrier of S
s1 is Neighbourhood of x0
N2 is Neighbourhood of x0
L is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
R is V16() V19( the carrier of T) V20( the carrier of S) Function-like (T,S) Element of K6(K7( the carrier of T, the carrier of S))
L is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
R is V16() V19( the carrier of T) V20( the carrier of S) Function-like (T,S) Element of K6(K7( the carrier of T, the carrier of S))
x is V11() real ext-real Element of REAL
{ b₁ where b₁ is Element of the carrier of T : not x <= ||.(b₁ - x0).|| } is set
R . 0 is Element of the carrier of T
lim R is Element of the carrier of T
lim L is Element of the carrier of T
lim (L + R) is Element of the carrier of T
(0. T) + x0 is Element of the carrier of T
y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
R ^\ y is V1() V16() V19( NAT ) V20( the carrier of T) Function-like constant total quasi_total subsequence of R
rng (R ^\ y) is trivial Element of K6( the carrier of T)
(L + R) ^\ y is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total subsequence of L + R
rng ((L + R) ^\ y) is Element of K6( the carrier of T)
n is set
n is set
y1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
((L + R) ^\ y) . y1 is Element of the carrier of T
y + 0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
y + y1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
(L + R) . (y + y1) is Element of the carrier of T
((L + R) . (y + y1)) - x0 is Element of the carrier of T
- x0 is Element of the carrier of T
K192(T,((L + R) . (y + y1)),(- x0)) is Element of the carrier of T
||.(((L + R) . (y + y1)) - x0).|| is V11() real ext-real Element of REAL
(((L + R) ^\ y) . y1) - x0 is Element of the carrier of T
K192(T,(((L + R) ^\ y) . y1),(- x0)) is Element of the carrier of T
||.((((L + R) ^\ y) . y1) - x0).|| is V11() real ext-real Element of REAL
y is Element of the carrier of T
y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
R ^\ y is V1() V16() V19( NAT ) V20( the carrier of T) Function-like constant total quasi_total subsequence of R
rng (R ^\ y) is trivial Element of K6( the carrier of T)
(L + R) ^\ y is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total subsequence of L + R
rng ((L + R) ^\ y) is Element of K6( the carrier of T)
y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
R ^\ y is V1() V16() V19( NAT ) V20( the carrier of T) Function-like constant total quasi_total subsequence of R
rng (R ^\ y) is trivial Element of K6( the carrier of T)
(L + R) ^\ y is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total subsequence of L + R
rng ((L + R) ^\ y) is Element of K6( the carrier of T)
L ^\ y is V1() V16() V19( NAT ) V20( the carrier of T) V20( the carrier of T) Function-like total quasi_total quasi_total non-zero (T, 0. T) subsequence of L
lim (L ^\ y) is Element of the carrier of T
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
((L + R) ^\ y) . n is Element of the carrier of T
f /. (((L + R) ^\ y) . n) is Element of the carrier of S
(R ^\ y) . n is Element of the carrier of T
f /. ((R ^\ y) . n) is Element of the carrier of S
(f /. (((L + R) ^\ y) . n)) - (f /. ((R ^\ y) . n)) is Element of the carrier of S
- (f /. ((R ^\ y) . n)) is Element of the carrier of S
K192(S,(f /. (((L + R) ^\ y) . n)),(- (f /. ((R ^\ y) . n)))) is Element of the carrier of S
(L ^\ y) . n is Element of the carrier of T
L . ((L ^\ y) . n) is Element of the carrier of S
R /. ((L ^\ y) . n) is Element of the carrier of S
(L . ((L ^\ y) . n)) + (R /. ((L ^\ y) . n)) is Element of the carrier of S
(((L + R) ^\ y) . n) - ((R ^\ y) . n) is Element of the carrier of T
- ((R ^\ y) . n) is Element of the carrier of T
K192(T,(((L + R) ^\ y) . n),(- ((R ^\ y) . n))) is Element of the carrier of T
n + y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
(L + R) . (n + y) is Element of the carrier of T
((L + R) . (n + y)) - ((R ^\ y) . n) is Element of the carrier of T
K192(T,((L + R) . (n + y)),(- ((R ^\ y) . n))) is Element of the carrier of T
L . (n + y) is Element of the carrier of T
R . (n + y) is Element of the carrier of T
(L . (n + y)) + (R . (n + y)) is Element of the carrier of T
((L . (n + y)) + (R . (n + y))) - ((R ^\ y) . n) is Element of the carrier of T
K192(T,((L . (n + y)) + (R . (n + y))),(- ((R ^\ y) . n))) is Element of the carrier of T
((L ^\ y) . n) + (R . (n + y)) is Element of the carrier of T
(((L ^\ y) . n) + (R . (n + y))) - ((R ^\ y) . n) is Element of the carrier of T
K192(T,(((L ^\ y) . n) + (R . (n + y))),(- ((R ^\ y) . n))) is Element of the carrier of T
((L ^\ y) . n) + ((R ^\ y) . n) is Element of the carrier of T
(((L ^\ y) . n) + ((R ^\ y) . n)) - ((R ^\ y) . n) is Element of the carrier of T
K192(T,(((L ^\ y) . n) + ((R ^\ y) . n)),(- ((R ^\ y) . n))) is Element of the carrier of T
((R ^\ y) . n) - ((R ^\ y) . n) is Element of the carrier of T
K192(T,((R ^\ y) . n),(- ((R ^\ y) . n))) is Element of the carrier of T
((L ^\ y) . n) + (((R ^\ y) . n) - ((R ^\ y) . n)) is Element of the carrier of T
((L ^\ y) . n) + (0. T) is Element of the carrier of T
BoundedLinearOperators (T,S) is V1() Element of K6( the carrier of (R_VectorSpace_of_LinearOperators (T,S)))
R_VectorSpace_of_LinearOperators (T,S) is non empty V87() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators (T,S)) is V1() set
K6( the carrier of (R_VectorSpace_of_LinearOperators (T,S))) is set
Zero_ ((BoundedLinearOperators (T,S)),(R_VectorSpace_of_LinearOperators (T,S))) is Element of BoundedLinearOperators (T,S)
Add_ ((BoundedLinearOperators (T,S)),(R_VectorSpace_of_LinearOperators (T,S))) is V16() V19(K7((BoundedLinearOperators (T,S)),(BoundedLinearOperators (T,S)))) V20( BoundedLinearOperators (T,S)) Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators (T,S)),(BoundedLinearOperators (T,S))),(BoundedLinearOperators (T,S))))
K7((BoundedLinearOperators (T,S)),(BoundedLinearOperators (T,S))) is set
K7(K7((BoundedLinearOperators (T,S)),(BoundedLinearOperators (T,S))),(BoundedLinearOperators (T,S))) is set
K6(K7(K7((BoundedLinearOperators (T,S)),(BoundedLinearOperators (T,S))),(BoundedLinearOperators (T,S)))) is set
Mult_ ((BoundedLinearOperators (T,S)),(R_VectorSpace_of_LinearOperators (T,S))) is V16() V19(K7(REAL,(BoundedLinearOperators (T,S)))) V20( BoundedLinearOperators (T,S)) Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators (T,S))),(BoundedLinearOperators (T,S))))
K7(REAL,(BoundedLinearOperators (T,S))) is set
K7(K7(REAL,(BoundedLinearOperators (T,S))),(BoundedLinearOperators (T,S))) is set
K6(K7(K7(REAL,(BoundedLinearOperators (T,S))),(BoundedLinearOperators (T,S)))) is set
BoundedLinearOperatorsNorm (T,S) is V1() V16() V19( BoundedLinearOperators (T,S)) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators (T,S)),REAL))
K7((BoundedLinearOperators (T,S)),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((BoundedLinearOperators (T,S)),REAL)) is set
NORMSTR(# (BoundedLinearOperators (T,S)),(Zero_ ((BoundedLinearOperators (T,S)),(R_VectorSpace_of_LinearOperators (T,S)))),(Add_ ((BoundedLinearOperators (T,S)),(R_VectorSpace_of_LinearOperators (T,S)))),(Mult_ ((BoundedLinearOperators (T,S)),(R_VectorSpace_of_LinearOperators (T,S)))),(BoundedLinearOperatorsNorm (T,S)) #) is strict NORMSTR
n is Element of BoundedLinearOperators (T,S)
modetrans (n,T,S) is V1() V16() V19( the carrier of T) V20( the carrier of S) Function-like total quasi_total V155(T,S) V156(T,S) Lipschitzian Element of K6(K7( the carrier of T, the carrier of S))
R /* (L ^\ y) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
lim (R /* (L ^\ y)) is Element of the carrier of S
y is set
dom R is Element of K6( the carrier of T)
rng (L ^\ y) is Element of K6( the carrier of T)
y is set
dom (modetrans (n,T,S)) is Element of K6( the carrier of T)
f /* ((L + R) ^\ y) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
f /* (R ^\ y) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
(f /* ((L + R) ^\ y)) - (f /* (R ^\ y)) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
((f /* ((L + R) ^\ y)) - (f /* (R ^\ y))) . y is Element of the carrier of S
(f /* ((L + R) ^\ y)) . y is Element of the carrier of S
(f /* (R ^\ y)) . y is Element of the carrier of S
((f /* ((L + R) ^\ y)) . y) - ((f /* (R ^\ y)) . y) is Element of the carrier of S
- ((f /* (R ^\ y)) . y) is Element of the carrier of S
K192(S,((f /* ((L + R) ^\ y)) . y),(- ((f /* (R ^\ y)) . y))) is Element of the carrier of S
((L + R) ^\ y) . y is Element of the carrier of T
f /. (((L + R) ^\ y) . y) is Element of the carrier of S
(f /. (((L + R) ^\ y) . y)) - ((f /* (R ^\ y)) . y) is Element of the carrier of S
K192(S,(f /. (((L + R) ^\ y) . y)),(- ((f /* (R ^\ y)) . y))) is Element of the carrier of S
(R ^\ y) . y is Element of the carrier of T
f /. ((R ^\ y) . y) is Element of the carrier of S
(f /. (((L + R) ^\ y) . y)) - (f /. ((R ^\ y) . y)) is Element of the carrier of S
- (f /. ((R ^\ y) . y)) is Element of the carrier of S
K192(S,(f /. (((L + R) ^\ y) . y)),(- (f /. ((R ^\ y) . y)))) is Element of the carrier of S
(L ^\ y) . y is Element of the carrier of T
n . ((L ^\ y) . y) is Element of the carrier of S
R /. ((L ^\ y) . y) is Element of the carrier of S
(n . ((L ^\ y) . y)) + (R /. ((L ^\ y) . y)) is Element of the carrier of S
(modetrans (n,T,S)) . ((L ^\ y) . y) is Element of the carrier of S
((modetrans (n,T,S)) . ((L ^\ y) . y)) + (R /. ((L ^\ y) . y)) is Element of the carrier of S
y1 is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
y1 /. ((L ^\ y) . y) is Element of the carrier of S
(y1 /. ((L ^\ y) . y)) + (R /. ((L ^\ y) . y)) is Element of the carrier of S
y1 /* (L ^\ y) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
(y1 /* (L ^\ y)) . y is Element of the carrier of S
((y1 /* (L ^\ y)) . y) + (R /. ((L ^\ y) . y)) is Element of the carrier of S
(R /* (L ^\ y)) . y is Element of the carrier of S
((y1 /* (L ^\ y)) . y) + ((R /* (L ^\ y)) . y) is Element of the carrier of S
(y1 /* (L ^\ y)) + (R /* (L ^\ y)) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
((y1 /* (L ^\ y)) + (R /* (L ^\ y))) . y is Element of the carrier of S
dom y1 is Element of K6( the carrier of T)
K is V11() real ext-real Element of REAL
K + 1 is V11() real ext-real Element of REAL
1 + K is V11() real ext-real Element of REAL
0 + K is V11() real ext-real Element of REAL
x1 is Element of the carrier of T
x2 is Element of the carrier of T
x1 - x2 is Element of the carrier of T
- x2 is Element of the carrier of T
K192(T,x1,(- x2)) is Element of the carrier of T
(modetrans (n,T,S)) . (x1 - x2) is Element of the carrier of S
||.((modetrans (n,T,S)) . (x1 - x2)).|| is V11() real ext-real Element of REAL
||.(x1 - x2).|| is V11() real ext-real Element of REAL
K * ||.(x1 - x2).|| is V11() real ext-real Element of REAL
(K + 1) * ||.(x1 - x2).|| is V11() real ext-real Element of REAL
y1 /. x1 is Element of the carrier of S
y1 /. x2 is Element of the carrier of S
(y1 /. x1) - (y1 /. x2) is Element of the carrier of S
- (y1 /. x2) is Element of the carrier of S
K192(S,(y1 /. x1),(- (y1 /. x2))) is Element of the carrier of S
||.((y1 /. x1) - (y1 /. x2)).|| is V11() real ext-real Element of REAL
(modetrans (n,T,S)) . x1 is Element of the carrier of S
((modetrans (n,T,S)) . x1) - (y1 /. x2) is Element of the carrier of S
K192(S,((modetrans (n,T,S)) . x1),(- (y1 /. x2))) is Element of the carrier of S
||.(((modetrans (n,T,S)) . x1) - (y1 /. x2)).|| is V11() real ext-real Element of REAL
(modetrans (n,T,S)) . x2 is Element of the carrier of S
- ((modetrans (n,T,S)) . x2) is Element of the carrier of S
((modetrans (n,T,S)) . x1) + (- ((modetrans (n,T,S)) . x2)) is Element of the carrier of S
||.(((modetrans (n,T,S)) . x1) + (- ((modetrans (n,T,S)) . x2))).|| is V11() real ext-real Element of REAL
(- 1) * ((modetrans (n,T,S)) . x2) is Element of the carrier of S
the Mult of S is V16() V19(K7(REAL, the carrier of S)) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of S), the carrier of S))
K7(REAL, the carrier of S) is set
K7(K7(REAL, the carrier of S), the carrier of S) is set
K6(K7(K7(REAL, the carrier of S), the carrier of S)) is set
K249( the Mult of S,(- 1),((modetrans (n,T,S)) . x2)) is set
((modetrans (n,T,S)) . x1) + ((- 1) * ((modetrans (n,T,S)) . x2)) is Element of the carrier of S
||.(((modetrans (n,T,S)) . x1) + ((- 1) * ((modetrans (n,T,S)) . x2))).|| is V11() real ext-real Element of REAL
(- 1) * x2 is Element of the carrier of T
the Mult of T is V16() V19(K7(REAL, the carrier of T)) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of T), the carrier of T))
K7(REAL, the carrier of T) is set
K7(K7(REAL, the carrier of T), the carrier of T) is set
K6(K7(K7(REAL, the carrier of T), the carrier of T)) is set
K249( the Mult of T,(- 1),x2) is set
(modetrans (n,T,S)) . ((- 1) * x2) is Element of the carrier of S
((modetrans (n,T,S)) . x1) + ((modetrans (n,T,S)) . ((- 1) * x2)) is Element of the carrier of S
||.(((modetrans (n,T,S)) . x1) + ((modetrans (n,T,S)) . ((- 1) * x2))).|| is V11() real ext-real Element of REAL
x1 + ((- 1) * x2) is Element of the carrier of T
(modetrans (n,T,S)) . (x1 + ((- 1) * x2)) is Element of the carrier of S
||.((modetrans (n,T,S)) . (x1 + ((- 1) * x2))).|| is V11() real ext-real Element of REAL
x1 is Element of the carrier of T
x2 is Element of the carrier of T
y1 /. x1 is Element of the carrier of S
y1 /. x2 is Element of the carrier of S
(y1 /. x1) - (y1 /. x2) is Element of the carrier of S
- (y1 /. x2) is Element of the carrier of S
K192(S,(y1 /. x1),(- (y1 /. x2))) is Element of the carrier of S
||.((y1 /. x1) - (y1 /. x2)).|| is V11() real ext-real Element of REAL
x1 - x2 is Element of the carrier of T
- x2 is Element of the carrier of T
K192(T,x1,(- x2)) is Element of the carrier of T
||.(x1 - x2).|| is V11() real ext-real Element of REAL
(K + 1) * ||.(x1 - x2).|| is V11() real ext-real Element of REAL
y is set
y1 /. (0. T) is Element of the carrier of S
(modetrans (n,T,S)) . (0. T) is Element of the carrier of S
0 * (0. T) is Element of the carrier of T
the Mult of T is V16() V19(K7(REAL, the carrier of T)) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of T), the carrier of T))
K7(REAL, the carrier of T) is set
K7(K7(REAL, the carrier of T), the carrier of T) is set
K6(K7(K7(REAL, the carrier of T), the carrier of T)) is set
K249( the Mult of T,0,(0. T)) is set
(modetrans (n,T,S)) . (0 * (0. T)) is Element of the carrier of S
0 * ((modetrans (n,T,S)) . (0. T)) is Element of the carrier of S
the Mult of S is V16() V19(K7(REAL, the carrier of S)) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of S), the carrier of S))
K7(REAL, the carrier of S) is set
K7(K7(REAL, the carrier of S), the carrier of S) is set
K6(K7(K7(REAL, the carrier of S), the carrier of S)) is set
K249( the Mult of S,0,((modetrans (n,T,S)) . (0. T))) is set
lim (y1 /* (L ^\ y)) is Element of the carrier of S
lim ((y1 /* (L ^\ y)) + (R /* (L ^\ y))) is Element of the carrier of S
(0. S) + (0. S) is Element of the carrier of S
y is set
(f /* (L + R)) ^\ y is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total subsequence of f /* (L + R)
((f /* (L + R)) ^\ y) - (f /* (R ^\ y)) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
(f /* R) ^\ y is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total subsequence of f /* R
((f /* (L + R)) ^\ y) - ((f /* R) ^\ y) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
((f /* (L + R)) - (f /* R)) ^\ y is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total subsequence of (f /* (L + R)) - (f /* R)
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
K7( the carrier of S, the carrier of T) is set
K6(K7( the carrier of S, the carrier of T)) is set
R_NormSpace_of_BoundedLinearOperators (S,T) is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
f is V16() V19( the carrier of S) V20( the carrier of T) Function-like Element of K6(K7( the carrier of S, the carrier of T))
x0 is V16() V19( the carrier of S) V20( the carrier of T) Function-like Element of K6(K7( the carrier of S, the carrier of T))
f + x0 is V16() V19( the carrier of S) V20( the carrier of T) Function-like Element of K6(K7( the carrier of S, the carrier of T))
N is Element of the carrier of S
(S,T,(f + x0),N) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T)) is V1() set
(S,T,f,N) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
(S,T,x0,N) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
(S,T,f,N) + (S,T,x0,N) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
dom f is Element of K6( the carrier of S)
K6( the carrier of S) is set
f /. N is Element of the carrier of T
L is Neighbourhood of N
R is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
s1 is V16() V19( the carrier of S) V20( the carrier of T) Function-like (S,T) Element of K6(K7( the carrier of S, the carrier of T))
R is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
s1 is V16() V19( the carrier of S) V20( the carrier of T) Function-like (S,T) Element of K6(K7( the carrier of S, the carrier of T))
dom x0 is Element of K6( the carrier of S)
x0 /. N is Element of the carrier of T
N2 is Neighbourhood of N
L is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
R is V16() V19( the carrier of S) V20( the carrier of T) Function-like (S,T) Element of K6(K7( the carrier of S, the carrier of T))
L is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
R is V16() V19( the carrier of S) V20( the carrier of T) Function-like (S,T) Element of K6(K7( the carrier of S, the carrier of T))
s1 + R is V16() V19( the carrier of S) V20( the carrier of T) Function-like Element of K6(K7( the carrier of S, the carrier of T))
R + L is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
n is Neighbourhood of N
n /\ n is Element of K6( the carrier of S)
(dom f) /\ (dom x0) is Element of K6( the carrier of S)
dom (f + x0) is Element of K6( the carrier of S)
y1 is Element of the carrier of S
(f + x0) /. y1 is Element of the carrier of T
(f + x0) /. N is Element of the carrier of T
((f + x0) /. y1) - ((f + x0) /. N) is Element of the carrier of T
- ((f + x0) /. N) is Element of the carrier of T
K192(T,((f + x0) /. y1),(- ((f + x0) /. N))) is Element of the carrier of T
f /. y1 is Element of the carrier of T
x0 /. y1 is Element of the carrier of T
(f /. y1) + (x0 /. y1) is Element of the carrier of T
((f /. y1) + (x0 /. y1)) - ((f + x0) /. N) is Element of the carrier of T
K192(T,((f /. y1) + (x0 /. y1)),(- ((f + x0) /. N))) is Element of the carrier of T
(f /. N) + (x0 /. N) is Element of the carrier of T
((f /. y1) + (x0 /. y1)) - ((f /. N) + (x0 /. N)) is Element of the carrier of T
- ((f /. N) + (x0 /. N)) is Element of the carrier of T
K192(T,((f /. y1) + (x0 /. y1)),(- ((f /. N) + (x0 /. N)))) is Element of the carrier of T
((f /. y1) + (x0 /. y1)) - (f /. N) is Element of the carrier of T
- (f /. N) is Element of the carrier of T
K192(T,((f /. y1) + (x0 /. y1)),(- (f /. N))) is Element of the carrier of T
(((f /. y1) + (x0 /. y1)) - (f /. N)) - (x0 /. N) is Element of the carrier of T
- (x0 /. N) is Element of the carrier of T
K192(T,(((f /. y1) + (x0 /. y1)) - (f /. N)),(- (x0 /. N))) is Element of the carrier of T
(f /. y1) + (- (f /. N)) is Element of the carrier of T
((f /. y1) + (- (f /. N))) + (x0 /. y1) is Element of the carrier of T
(((f /. y1) + (- (f /. N))) + (x0 /. y1)) - (x0 /. N) is Element of the carrier of T
K192(T,(((f /. y1) + (- (f /. N))) + (x0 /. y1)),(- (x0 /. N))) is Element of the carrier of T
(f /. y1) - (f /. N) is Element of the carrier of T
K192(T,(f /. y1),(- (f /. N))) is Element of the carrier of T
(x0 /. y1) - (x0 /. N) is Element of the carrier of T
K192(T,(x0 /. y1),(- (x0 /. N))) is Element of the carrier of T
((f /. y1) - (f /. N)) + ((x0 /. y1) - (x0 /. N)) is Element of the carrier of T
y1 - N is Element of the carrier of S
- N is Element of the carrier of S
K192(S,y1,(- N)) is Element of the carrier of S
R . (y1 - N) is Element of the carrier of T
s1 /. (y1 - N) is Element of the carrier of T
(R . (y1 - N)) + (s1 /. (y1 - N)) is Element of the carrier of T
((R . (y1 - N)) + (s1 /. (y1 - N))) + ((x0 /. y1) - (x0 /. N)) is Element of the carrier of T
L . (y1 - N) is Element of the carrier of T
R /. (y1 - N) is Element of the carrier of T
(L . (y1 - N)) + (R /. (y1 - N)) is Element of the carrier of T
((R . (y1 - N)) + (s1 /. (y1 - N))) + ((L . (y1 - N)) + (R /. (y1 - N))) is Element of the carrier of T
(s1 /. (y1 - N)) + (R . (y1 - N)) is Element of the carrier of T
((s1 /. (y1 - N)) + (R . (y1 - N))) + (L . (y1 - N)) is Element of the carrier of T
(((s1 /. (y1 - N)) + (R . (y1 - N))) + (L . (y1 - N))) + (R /. (y1 - N)) is Element of the carrier of T
(R . (y1 - N)) + (L . (y1 - N)) is Element of the carrier of T
((R . (y1 - N)) + (L . (y1 - N))) + (s1 /. (y1 - N)) is Element of the carrier of T
(((R . (y1 - N)) + (L . (y1 - N))) + (s1 /. (y1 - N))) + (R /. (y1 - N)) is Element of the carrier of T
(s1 /. (y1 - N)) + (R /. (y1 - N)) is Element of the carrier of T
((R . (y1 - N)) + (L . (y1 - N))) + ((s1 /. (y1 - N)) + (R /. (y1 - N))) is Element of the carrier of T
(R + L) . (y1 - N) is Element of the carrier of T
((R + L) . (y1 - N)) + ((s1 /. (y1 - N)) + (R /. (y1 - N))) is Element of the carrier of T
x is V16() V19( the carrier of S) V20( the carrier of T) Function-like (S,T) Element of K6(K7( the carrier of S, the carrier of T))
x /. (y1 - N) is Element of the carrier of T
((R + L) . (y1 - N)) + (x /. (y1 - N)) is Element of the carrier of T
(S,T,f,N) + L is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
K7( the carrier of S, the carrier of T) is set
K6(K7( the carrier of S, the carrier of T)) is set
R_NormSpace_of_BoundedLinearOperators (S,T) is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
f is V16() V19( the carrier of S) V20( the carrier of T) Function-like Element of K6(K7( the carrier of S, the carrier of T))
x0 is V16() V19( the carrier of S) V20( the carrier of T) Function-like Element of K6(K7( the carrier of S, the carrier of T))
f - x0 is V16() V19( the carrier of S) V20( the carrier of T) Function-like Element of K6(K7( the carrier of S, the carrier of T))
N is Element of the carrier of S
(S,T,(f - x0),N) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T)) is V1() set
(S,T,f,N) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
(S,T,x0,N) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
(S,T,f,N) - (S,T,x0,N) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
- (S,T,x0,N) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
K192((R_NormSpace_of_BoundedLinearOperators (S,T)),(S,T,f,N),(- (S,T,x0,N))) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
dom f is Element of K6( the carrier of S)
K6( the carrier of S) is set
f /. N is Element of the carrier of T
L is Neighbourhood of N
R is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
s1 is V16() V19( the carrier of S) V20( the carrier of T) Function-like (S,T) Element of K6(K7( the carrier of S, the carrier of T))
R is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
s1 is V16() V19( the carrier of S) V20( the carrier of T) Function-like (S,T) Element of K6(K7( the carrier of S, the carrier of T))
dom x0 is Element of K6( the carrier of S)
x0 /. N is Element of the carrier of T
N2 is Neighbourhood of N
L is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
R is V16() V19( the carrier of S) V20( the carrier of T) Function-like (S,T) Element of K6(K7( the carrier of S, the carrier of T))
L is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
R is V16() V19( the carrier of S) V20( the carrier of T) Function-like (S,T) Element of K6(K7( the carrier of S, the carrier of T))
s1 - R is V16() V19( the carrier of S) V20( the carrier of T) Function-like Element of K6(K7( the carrier of S, the carrier of T))
R - L is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
- L is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
K192((R_NormSpace_of_BoundedLinearOperators (S,T)),R,(- L)) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
n is Neighbourhood of N
n /\ n is Element of K6( the carrier of S)
(dom f) /\ (dom x0) is Element of K6( the carrier of S)
dom (f - x0) is Element of K6( the carrier of S)
y1 is Element of the carrier of S
(f - x0) /. y1 is Element of the carrier of T
(f - x0) /. N is Element of the carrier of T
((f - x0) /. y1) - ((f - x0) /. N) is Element of the carrier of T
- ((f - x0) /. N) is Element of the carrier of T
K192(T,((f - x0) /. y1),(- ((f - x0) /. N))) is Element of the carrier of T
f /. y1 is Element of the carrier of T
x0 /. y1 is Element of the carrier of T
(f /. y1) - (x0 /. y1) is Element of the carrier of T
- (x0 /. y1) is Element of the carrier of T
K192(T,(f /. y1),(- (x0 /. y1))) is Element of the carrier of T
((f /. y1) - (x0 /. y1)) - ((f - x0) /. N) is Element of the carrier of T
K192(T,((f /. y1) - (x0 /. y1)),(- ((f - x0) /. N))) is Element of the carrier of T
(f /. N) - (x0 /. N) is Element of the carrier of T
- (x0 /. N) is Element of the carrier of T
K192(T,(f /. N),(- (x0 /. N))) is Element of the carrier of T
((f /. y1) - (x0 /. y1)) - ((f /. N) - (x0 /. N)) is Element of the carrier of T
- ((f /. N) - (x0 /. N)) is Element of the carrier of T
K192(T,((f /. y1) - (x0 /. y1)),(- ((f /. N) - (x0 /. N)))) is Element of the carrier of T
((f /. y1) - (x0 /. y1)) - (f /. N) is Element of the carrier of T
- (f /. N) is Element of the carrier of T
K192(T,((f /. y1) - (x0 /. y1)),(- (f /. N))) is Element of the carrier of T
(((f /. y1) - (x0 /. y1)) - (f /. N)) + (x0 /. N) is Element of the carrier of T
(f /. N) + (x0 /. y1) is Element of the carrier of T
(f /. y1) - ((f /. N) + (x0 /. y1)) is Element of the carrier of T
- ((f /. N) + (x0 /. y1)) is Element of the carrier of T
K192(T,(f /. y1),(- ((f /. N) + (x0 /. y1)))) is Element of the carrier of T
((f /. y1) - ((f /. N) + (x0 /. y1))) + (x0 /. N) is Element of the carrier of T
(f /. y1) - (f /. N) is Element of the carrier of T
K192(T,(f /. y1),(- (f /. N))) is Element of the carrier of T
((f /. y1) - (f /. N)) - (x0 /. y1) is Element of the carrier of T
K192(T,((f /. y1) - (f /. N)),(- (x0 /. y1))) is Element of the carrier of T
(((f /. y1) - (f /. N)) - (x0 /. y1)) + (x0 /. N) is Element of the carrier of T
(x0 /. y1) - (x0 /. N) is Element of the carrier of T
K192(T,(x0 /. y1),(- (x0 /. N))) is Element of the carrier of T
((f /. y1) - (f /. N)) - ((x0 /. y1) - (x0 /. N)) is Element of the carrier of T
- ((x0 /. y1) - (x0 /. N)) is Element of the carrier of T
K192(T,((f /. y1) - (f /. N)),(- ((x0 /. y1) - (x0 /. N)))) is Element of the carrier of T
y1 - N is Element of the carrier of S
- N is Element of the carrier of S
K192(S,y1,(- N)) is Element of the carrier of S
R . (y1 - N) is Element of the carrier of T
s1 /. (y1 - N) is Element of the carrier of T
(R . (y1 - N)) + (s1 /. (y1 - N)) is Element of the carrier of T
((R . (y1 - N)) + (s1 /. (y1 - N))) - ((x0 /. y1) - (x0 /. N)) is Element of the carrier of T
K192(T,((R . (y1 - N)) + (s1 /. (y1 - N))),(- ((x0 /. y1) - (x0 /. N)))) is Element of the carrier of T
L . (y1 - N) is Element of the carrier of T
R /. (y1 - N) is Element of the carrier of T
(L . (y1 - N)) + (R /. (y1 - N)) is Element of the carrier of T
((R . (y1 - N)) + (s1 /. (y1 - N))) - ((L . (y1 - N)) + (R /. (y1 - N))) is Element of the carrier of T
- ((L . (y1 - N)) + (R /. (y1 - N))) is Element of the carrier of T
K192(T,((R . (y1 - N)) + (s1 /. (y1 - N))),(- ((L . (y1 - N)) + (R /. (y1 - N))))) is Element of the carrier of T
((R . (y1 - N)) + (s1 /. (y1 - N))) - (L . (y1 - N)) is Element of the carrier of T
- (L . (y1 - N)) is Element of the carrier of T
K192(T,((R . (y1 - N)) + (s1 /. (y1 - N))),(- (L . (y1 - N)))) is Element of the carrier of T
(((R . (y1 - N)) + (s1 /. (y1 - N))) - (L . (y1 - N))) - (R /. (y1 - N)) is Element of the carrier of T
- (R /. (y1 - N)) is Element of the carrier of T
K192(T,(((R . (y1 - N)) + (s1 /. (y1 - N))) - (L . (y1 - N))),(- (R /. (y1 - N)))) is Element of the carrier of T
(s1 /. (y1 - N)) + (- (L . (y1 - N))) is Element of the carrier of T
(R . (y1 - N)) + ((s1 /. (y1 - N)) + (- (L . (y1 - N)))) is Element of the carrier of T
((R . (y1 - N)) + ((s1 /. (y1 - N)) + (- (L . (y1 - N))))) - (R /. (y1 - N)) is Element of the carrier of T
K192(T,((R . (y1 - N)) + ((s1 /. (y1 - N)) + (- (L . (y1 - N))))),(- (R /. (y1 - N)))) is Element of the carrier of T
(L . (y1 - N)) - (s1 /. (y1 - N)) is Element of the carrier of T
- (s1 /. (y1 - N)) is Element of the carrier of T
K192(T,(L . (y1 - N)),(- (s1 /. (y1 - N)))) is Element of the carrier of T
(R . (y1 - N)) - ((L . (y1 - N)) - (s1 /. (y1 - N))) is Element of the carrier of T
- ((L . (y1 - N)) - (s1 /. (y1 - N))) is Element of the carrier of T
K192(T,(R . (y1 - N)),(- ((L . (y1 - N)) - (s1 /. (y1 - N))))) is Element of the carrier of T
((R . (y1 - N)) - ((L . (y1 - N)) - (s1 /. (y1 - N)))) - (R /. (y1 - N)) is Element of the carrier of T
K192(T,((R . (y1 - N)) - ((L . (y1 - N)) - (s1 /. (y1 - N)))),(- (R /. (y1 - N)))) is Element of the carrier of T
(R . (y1 - N)) - (L . (y1 - N)) is Element of the carrier of T
K192(T,(R . (y1 - N)),(- (L . (y1 - N)))) is Element of the carrier of T
((R . (y1 - N)) - (L . (y1 - N))) + (s1 /. (y1 - N)) is Element of the carrier of T
(((R . (y1 - N)) - (L . (y1 - N))) + (s1 /. (y1 - N))) - (R /. (y1 - N)) is Element of the carrier of T
K192(T,(((R . (y1 - N)) - (L . (y1 - N))) + (s1 /. (y1 - N))),(- (R /. (y1 - N)))) is Element of the carrier of T
(s1 /. (y1 - N)) - (R /. (y1 - N)) is Element of the carrier of T
K192(T,(s1 /. (y1 - N)),(- (R /. (y1 - N)))) is Element of the carrier of T
((R . (y1 - N)) - (L . (y1 - N))) + ((s1 /. (y1 - N)) - (R /. (y1 - N))) is Element of the carrier of T
(R - L) . (y1 - N) is Element of the carrier of T
((R - L) . (y1 - N)) + ((s1 /. (y1 - N)) - (R /. (y1 - N))) is Element of the carrier of T
x is V16() V19( the carrier of S) V20( the carrier of T) Function-like (S,T) Element of K6(K7( the carrier of S, the carrier of T))
x /. (y1 - N) is Element of the carrier of T
((R - L) . (y1 - N)) + (x /. (y1 - N)) is Element of the carrier of T
(S,T,f,N) - L is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
K192((R_NormSpace_of_BoundedLinearOperators (S,T)),(S,T,f,N),(- L)) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
K7( the carrier of S, the carrier of T) is set
K6(K7( the carrier of S, the carrier of T)) is set
R_NormSpace_of_BoundedLinearOperators (S,T) is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
f is V11() real ext-real Element of REAL
x0 is V16() V19( the carrier of S) V20( the carrier of T) Function-like Element of K6(K7( the carrier of S, the carrier of T))
f (#) x0 is V16() V19( the carrier of S) V20( the carrier of T) Function-like Element of K6(K7( the carrier of S, the carrier of T))
N is Element of the carrier of S
(S,T,(f (#) x0),N) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T)) is V1() set
(S,T,x0,N) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
f * (S,T,x0,N) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
the Mult of (R_NormSpace_of_BoundedLinearOperators (S,T)) is V16() V19(K7(REAL, the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T)))) V20( the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))), the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))))
K7(REAL, the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))) is set
K7(K7(REAL, the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))), the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))) is set
K6(K7(K7(REAL, the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))), the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T)))) is set
K249( the Mult of (R_NormSpace_of_BoundedLinearOperators (S,T)),f,(S,T,x0,N)) is set
dom x0 is Element of K6( the carrier of S)
K6( the carrier of S) is set
x0 /. N is Element of the carrier of T
L is Neighbourhood of N
R is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
s1 is V16() V19( the carrier of S) V20( the carrier of T) Function-like (S,T) Element of K6(K7( the carrier of S, the carrier of T))
R is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
s1 is V16() V19( the carrier of S) V20( the carrier of T) Function-like (S,T) Element of K6(K7( the carrier of S, the carrier of T))
f (#) s1 is V16() V19( the carrier of S) V20( the carrier of T) Function-like Element of K6(K7( the carrier of S, the carrier of T))
f * R is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
K249( the Mult of (R_NormSpace_of_BoundedLinearOperators (S,T)),f,R) is set
dom (f (#) x0) is Element of K6( the carrier of S)
R is Element of the carrier of S
(f (#) x0) /. R is Element of the carrier of T
(f (#) x0) /. N is Element of the carrier of T
((f (#) x0) /. R) - ((f (#) x0) /. N) is Element of the carrier of T
- ((f (#) x0) /. N) is Element of the carrier of T
K192(T,((f (#) x0) /. R),(- ((f (#) x0) /. N))) is Element of the carrier of T
x0 /. R is Element of the carrier of T
f * (x0 /. R) is Element of the carrier of T
the Mult of T is V16() V19(K7(REAL, the carrier of T)) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of T), the carrier of T))
K7(REAL, the carrier of T) is set
K7(K7(REAL, the carrier of T), the carrier of T) is set
K6(K7(K7(REAL, the carrier of T), the carrier of T)) is set
K249( the Mult of T,f,(x0 /. R)) is set
(f * (x0 /. R)) - ((f (#) x0) /. N) is Element of the carrier of T
K192(T,(f * (x0 /. R)),(- ((f (#) x0) /. N))) is Element of the carrier of T
f * (x0 /. N) is Element of the carrier of T
K249( the Mult of T,f,(x0 /. N)) is set
(f * (x0 /. R)) - (f * (x0 /. N)) is Element of the carrier of T
- (f * (x0 /. N)) is Element of the carrier of T
K192(T,(f * (x0 /. R)),(- (f * (x0 /. N)))) is Element of the carrier of T
(x0 /. R) - (x0 /. N) is Element of the carrier of T
- (x0 /. N) is Element of the carrier of T
K192(T,(x0 /. R),(- (x0 /. N))) is Element of the carrier of T
f * ((x0 /. R) - (x0 /. N)) is Element of the carrier of T
K249( the Mult of T,f,((x0 /. R) - (x0 /. N))) is set
R - N is Element of the carrier of S
- N is Element of the carrier of S
K192(S,R,(- N)) is Element of the carrier of S
R . (R - N) is Element of the carrier of T
s1 /. (R - N) is Element of the carrier of T
(R . (R - N)) + (s1 /. (R - N)) is Element of the carrier of T
f * ((R . (R - N)) + (s1 /. (R - N))) is Element of the carrier of T
K249( the Mult of T,f,((R . (R - N)) + (s1 /. (R - N)))) is set
f * (R . (R - N)) is Element of the carrier of T
K249( the Mult of T,f,(R . (R - N))) is set
f * (s1 /. (R - N)) is Element of the carrier of T
K249( the Mult of T,f,(s1 /. (R - N))) is set
(f * (R . (R - N))) + (f * (s1 /. (R - N))) is Element of the carrier of T
(f * R) . (R - N) is Element of the carrier of T
((f * R) . (R - N)) + (f * (s1 /. (R - N))) is Element of the carrier of T
N2 is V16() V19( the carrier of S) V20( the carrier of T) Function-like (S,T) Element of K6(K7( the carrier of S, the carrier of T))
N2 /. (R - N) is Element of the carrier of T
((f * R) . (R - N)) + (N2 /. (R - N)) is Element of the carrier of T
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
K7( the carrier of T, the carrier of T) is set
K6(K7( the carrier of T, the carrier of T)) is set
K6( the carrier of T) is set
R_NormSpace_of_BoundedLinearOperators (T,T) is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators (T,T)) is V1() set
id the carrier of T is V1() V16() V19( the carrier of T) V20( the carrier of T) Function-like one-to-one total quasi_total Element of K6(K7( the carrier of T, the carrier of T))
BoundedLinearOperators (T,T) is V1() Element of K6( the carrier of (R_VectorSpace_of_LinearOperators (T,T)))
R_VectorSpace_of_LinearOperators (T,T) is non empty V87() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators (T,T)) is V1() set
K6( the carrier of (R_VectorSpace_of_LinearOperators (T,T))) is set
Zero_ ((BoundedLinearOperators (T,T)),(R_VectorSpace_of_LinearOperators (T,T))) is Element of BoundedLinearOperators (T,T)
Add_ ((BoundedLinearOperators (T,T)),(R_VectorSpace_of_LinearOperators (T,T))) is V16() V19(K7((BoundedLinearOperators (T,T)),(BoundedLinearOperators (T,T)))) V20( BoundedLinearOperators (T,T)) Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators (T,T)),(BoundedLinearOperators (T,T))),(BoundedLinearOperators (T,T))))
K7((BoundedLinearOperators (T,T)),(BoundedLinearOperators (T,T))) is set
K7(K7((BoundedLinearOperators (T,T)),(BoundedLinearOperators (T,T))),(BoundedLinearOperators (T,T))) is set
K6(K7(K7((BoundedLinearOperators (T,T)),(BoundedLinearOperators (T,T))),(BoundedLinearOperators (T,T)))) is set
Mult_ ((BoundedLinearOperators (T,T)),(R_VectorSpace_of_LinearOperators (T,T))) is V16() V19(K7(REAL,(BoundedLinearOperators (T,T)))) V20( BoundedLinearOperators (T,T)) Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators (T,T))),(BoundedLinearOperators (T,T))))
K7(REAL,(BoundedLinearOperators (T,T))) is set
K7(K7(REAL,(BoundedLinearOperators (T,T))),(BoundedLinearOperators (T,T))) is set
K6(K7(K7(REAL,(BoundedLinearOperators (T,T))),(BoundedLinearOperators (T,T)))) is set
BoundedLinearOperatorsNorm (T,T) is V1() V16() V19( BoundedLinearOperators (T,T)) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators (T,T)),REAL))
K7((BoundedLinearOperators (T,T)),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((BoundedLinearOperators (T,T)),REAL)) is set
NORMSTR(# (BoundedLinearOperators (T,T)),(Zero_ ((BoundedLinearOperators (T,T)),(R_VectorSpace_of_LinearOperators (T,T)))),(Add_ ((BoundedLinearOperators (T,T)),(R_VectorSpace_of_LinearOperators (T,T)))),(Mult_ ((BoundedLinearOperators (T,T)),(R_VectorSpace_of_LinearOperators (T,T)))),(BoundedLinearOperatorsNorm (T,T)) #) is strict NORMSTR
x0 is V16() V19( the carrier of T) V20( the carrier of T) Function-like Element of K6(K7( the carrier of T, the carrier of T))
dom x0 is Element of K6( the carrier of T)
N is Element of K6( the carrier of T)
x0 | N is V16() V19( the carrier of T) V20( the carrier of T) Function-like Element of K6(K7( the carrier of T, the carrier of T))
id N is V16() V19(N) V20(N) Function-like one-to-one total quasi_total Element of K6(K7(N,N))
K7(N,N) is set
K6(K7(N,N)) is set
(T,T,x0,N) is V16() V19( the carrier of T) V20( the carrier of (R_NormSpace_of_BoundedLinearOperators (T,T))) Function-like Element of K6(K7( the carrier of T, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,T))))
K7( the carrier of T, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,T))) is set
K6(K7( the carrier of T, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,T)))) is set
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
the carrier of T --> (0. T) is V1() V16() V19( the carrier of T) V20( the carrier of T) Function-like constant total quasi_total Element of K6(K7( the carrier of T, the carrier of T))
L is V16() V19( the carrier of T) V20( the carrier of T) Function-like Element of K6(K7( the carrier of T, the carrier of T))
dom L is Element of K6( the carrier of T)
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
R is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total non-zero (T, 0. T) Element of K6(K7(NAT, the carrier of T))
s1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
R . s1 is Element of the carrier of T
L /. (R . s1) is Element of the carrier of T
L . (R . s1) is set
rng R is Element of K6( the carrier of T)
L /* R is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
||.R.|| is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
||.R.|| " is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(T,(L /* R),(||.R.|| ")) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(T,(L /* R),(||.R.|| ")) . s1 is Element of the carrier of T
(L /* R) . s1 is Element of the carrier of T
(||.R.|| ") . s1 is V11() real ext-real Element of REAL
((||.R.|| ") . s1) * ((L /* R) . s1) is Element of the carrier of T
the Mult of T is V16() V19(K7(REAL, the carrier of T)) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of T), the carrier of T))
K7(REAL, the carrier of T) is set
K7(K7(REAL, the carrier of T), the carrier of T) is set
K6(K7(K7(REAL, the carrier of T), the carrier of T)) is set
K249( the Mult of T,((||.R.|| ") . s1),((L /* R) . s1)) is set
((||.R.|| ") . s1) * (L /. (R . s1)) is Element of the carrier of T
K249( the Mult of T,((||.R.|| ") . s1),(L /. (R . s1))) is set
(T,(L /* R),(||.R.|| ")) . 0 is Element of the carrier of T
lim (T,(L /* R),(||.R.|| ")) is Element of the carrier of T
s1 is Element of the carrier of T
(x0 | N) . s1 is set
x0 . s1 is set
x0 /. s1 is Element of the carrier of T
s1 is Element of the carrier of T
N2 is Neighbourhood of s1
x0 /. s1 is Element of the carrier of T
f is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,T))
R is V16() V19( the carrier of T) V20( the carrier of T) Function-like (T,T) Element of K6(K7( the carrier of T, the carrier of T))
L is Element of the carrier of T
x0 /. L is Element of the carrier of T
(x0 /. L) - (x0 /. s1) is Element of the carrier of T
- (x0 /. s1) is Element of the carrier of T
K192(T,(x0 /. L),(- (x0 /. s1))) is Element of the carrier of T
L - s1 is Element of the carrier of T
- s1 is Element of the carrier of T
K192(T,L,(- s1)) is Element of the carrier of T
f . (L - s1) is Element of the carrier of T
R /. (L - s1) is Element of the carrier of T
(f . (L - s1)) + (R /. (L - s1)) is Element of the carrier of T
R . (L - s1) is set
L - (x0 /. s1) is Element of the carrier of T
K192(T,L,(- (x0 /. s1))) is Element of the carrier of T
s1 is Element of the carrier of T
(T,T,x0,N) /. s1 is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,T))
N2 is Neighbourhood of s1
x0 /. s1 is Element of the carrier of T
L is Neighbourhood of s1
R is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,T))
x is V16() V19( the carrier of T) V20( the carrier of T) Function-like (T,T) Element of K6(K7( the carrier of T, the carrier of T))
L is Neighbourhood of s1
R is Neighbourhood of s1
x is Element of the carrier of T
x0 /. x is Element of the carrier of T
(x0 /. x) - (x0 /. s1) is Element of the carrier of T
- (x0 /. s1) is Element of the carrier of T
K192(T,(x0 /. x),(- (x0 /. s1))) is Element of the carrier of T
x - s1 is Element of the carrier of T
- s1 is Element of the carrier of T
K192(T,x,(- s1)) is Element of the carrier of T
f . (x - s1) is Element of the carrier of T
R /. (x - s1) is Element of the carrier of T
(f . (x - s1)) + (R /. (x - s1)) is Element of the carrier of T
R . (x - s1) is set
x - (x0 /. s1) is Element of the carrier of T
K192(T,x,(- (x0 /. s1))) is Element of the carrier of T
(T,T,x0,s1) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,T))
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
K6( the carrier of T) is set
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
K7( the carrier of T, the carrier of S) is set
K6(K7( the carrier of T, the carrier of S)) is set
R_NormSpace_of_BoundedLinearOperators (T,S) is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S)) is V1() set
f is Element of K6( the carrier of T)
x0 is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
N is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
x0 + N is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
dom (x0 + N) is Element of K6( the carrier of T)
(T,S,(x0 + N),f) is V16() V19( the carrier of T) V20( the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))) Function-like Element of K6(K7( the carrier of T, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))))
K7( the carrier of T, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))) is set
K6(K7( the carrier of T, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S)))) is set
L is Element of the carrier of T
L is Element of the carrier of T
(T,S,(x0 + N),f) /. L is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
(T,S,(x0 + N),L) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
(T,S,x0,L) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
(T,S,N,L) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
(T,S,x0,L) + (T,S,N,L) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
L is Element of the carrier of T
(T,S,(x0 + N),f) /. L is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
(T,S,x0,L) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
(T,S,N,L) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
(T,S,x0,L) + (T,S,N,L) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
K6( the carrier of T) is set
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
K7( the carrier of T, the carrier of S) is set
K6(K7( the carrier of T, the carrier of S)) is set
R_NormSpace_of_BoundedLinearOperators (T,S) is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S)) is V1() set
f is Element of K6( the carrier of T)
x0 is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
N is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
x0 - N is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
dom (x0 - N) is Element of K6( the carrier of T)
(T,S,(x0 - N),f) is V16() V19( the carrier of T) V20( the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))) Function-like Element of K6(K7( the carrier of T, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))))
K7( the carrier of T, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))) is set
K6(K7( the carrier of T, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S)))) is set
L is Element of the carrier of T
L is Element of the carrier of T
(T,S,(x0 - N),f) /. L is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
(T,S,(x0 - N),L) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
(T,S,x0,L) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
(T,S,N,L) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
(T,S,x0,L) - (T,S,N,L) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
- (T,S,N,L) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
K192((R_NormSpace_of_BoundedLinearOperators (T,S)),(T,S,x0,L),(- (T,S,N,L))) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
L is Element of the carrier of T
(T,S,(x0 - N),f) /. L is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
(T,S,x0,L) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
(T,S,N,L) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
(T,S,x0,L) - (T,S,N,L) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
- (T,S,N,L) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
K192((R_NormSpace_of_BoundedLinearOperators (T,S)),(T,S,x0,L),(- (T,S,N,L))) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
K6( the carrier of T) is set
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
K7( the carrier of T, the carrier of S) is set
K6(K7( the carrier of T, the carrier of S)) is set
R_NormSpace_of_BoundedLinearOperators (T,S) is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S)) is V1() set
f is Element of K6( the carrier of T)
x0 is V11() real ext-real Element of REAL
N is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
x0 (#) N is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
dom (x0 (#) N) is Element of K6( the carrier of T)
(T,S,(x0 (#) N),f) is V16() V19( the carrier of T) V20( the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))) Function-like Element of K6(K7( the carrier of T, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))))
K7( the carrier of T, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))) is set
K6(K7( the carrier of T, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S)))) is set
L is Element of the carrier of T
L is Element of the carrier of T
(T,S,(x0 (#) N),f) /. L is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
(T,S,(x0 (#) N),L) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
(T,S,N,L) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
x0 * (T,S,N,L) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
the Mult of (R_NormSpace_of_BoundedLinearOperators (T,S)) is V16() V19(K7(REAL, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S)))) V20( the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))), the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))))
K7(REAL, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))) is set
K7(K7(REAL, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))), the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))) is set
K6(K7(K7(REAL, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))), the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S)))) is set
K249( the Mult of (R_NormSpace_of_BoundedLinearOperators (T,S)),x0,(T,S,N,L)) is set
L is Element of the carrier of T
(T,S,(x0 (#) N),f) /. L is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
(T,S,N,L) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
x0 * (T,S,N,L) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
K249( the Mult of (R_NormSpace_of_BoundedLinearOperators (T,S)),x0,(T,S,N,L)) is set
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
K6( the carrier of T) is set
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
K7( the carrier of T, the carrier of S) is set
K6(K7( the carrier of T, the carrier of S)) is set
R_NormSpace_of_BoundedLinearOperators (T,S) is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S)) is V1() set
0. (R_NormSpace_of_BoundedLinearOperators (T,S)) is V16() Function-like V68( R_NormSpace_of_BoundedLinearOperators (T,S)) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
the ZeroF of (R_NormSpace_of_BoundedLinearOperators (T,S)) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
f is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
dom f is Element of K6( the carrier of T)
x0 is Element of K6( the carrier of T)
f | x0 is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
(T,S,f,x0) is V16() V19( the carrier of T) V20( the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))) Function-like Element of K6(K7( the carrier of T, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))))
K7( the carrier of T, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))) is set
K6(K7( the carrier of T, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S)))) is set
0. S is V68(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
the carrier of T --> (0. S) is V1() V16() V19( the carrier of T) V20( the carrier of S) Function-like constant total quasi_total Element of K6(K7( the carrier of T, the carrier of S))
BoundedLinearOperators (T,S) is V1() Element of K6( the carrier of (R_VectorSpace_of_LinearOperators (T,S)))
R_VectorSpace_of_LinearOperators (T,S) is non empty V87() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators (T,S)) is V1() set
K6( the carrier of (R_VectorSpace_of_LinearOperators (T,S))) is set
Zero_ ((BoundedLinearOperators (T,S)),(R_VectorSpace_of_LinearOperators (T,S))) is Element of BoundedLinearOperators (T,S)
Add_ ((BoundedLinearOperators (T,S)),(R_VectorSpace_of_LinearOperators (T,S))) is V16() V19(K7((BoundedLinearOperators (T,S)),(BoundedLinearOperators (T,S)))) V20( BoundedLinearOperators (T,S)) Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators (T,S)),(BoundedLinearOperators (T,S))),(BoundedLinearOperators (T,S))))
K7((BoundedLinearOperators (T,S)),(BoundedLinearOperators (T,S))) is set
K7(K7((BoundedLinearOperators (T,S)),(BoundedLinearOperators (T,S))),(BoundedLinearOperators (T,S))) is set
K6(K7(K7((BoundedLinearOperators (T,S)),(BoundedLinearOperators (T,S))),(BoundedLinearOperators (T,S)))) is set
Mult_ ((BoundedLinearOperators (T,S)),(R_VectorSpace_of_LinearOperators (T,S))) is V16() V19(K7(REAL,(BoundedLinearOperators (T,S)))) V20( BoundedLinearOperators (T,S)) Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators (T,S))),(BoundedLinearOperators (T,S))))
K7(REAL,(BoundedLinearOperators (T,S))) is set
K7(K7(REAL,(BoundedLinearOperators (T,S))),(BoundedLinearOperators (T,S))) is set
K6(K7(K7(REAL,(BoundedLinearOperators (T,S))),(BoundedLinearOperators (T,S)))) is set
BoundedLinearOperatorsNorm (T,S) is V1() V16() V19( BoundedLinearOperators (T,S)) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators (T,S)),REAL))
K7((BoundedLinearOperators (T,S)),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((BoundedLinearOperators (T,S)),REAL)) is set
NORMSTR(# (BoundedLinearOperators (T,S)),(Zero_ ((BoundedLinearOperators (T,S)),(R_VectorSpace_of_LinearOperators (T,S)))),(Add_ ((BoundedLinearOperators (T,S)),(R_VectorSpace_of_LinearOperators (T,S)))),(Mult_ ((BoundedLinearOperators (T,S)),(R_VectorSpace_of_LinearOperators (T,S)))),(BoundedLinearOperatorsNorm (T,S)) #) is strict NORMSTR
N is V16() V19( the carrier of T) V20( the carrier of S) Function-like Element of K6(K7( the carrier of T, the carrier of S))
dom N is Element of K6( the carrier of T)
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
s1 is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total non-zero (T, 0. T) Element of K6(K7(NAT, the carrier of T))
N2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
s1 . N2 is Element of the carrier of T
N /. (s1 . N2) is Element of the carrier of S
N . (s1 . N2) is set
rng s1 is Element of K6( the carrier of T)
N /* s1 is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
K7(NAT, the carrier of S) is set
K6(K7(NAT, the carrier of S)) is set
||.s1.|| is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
||.s1.|| " is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(S,(N /* s1),(||.s1.|| ")) is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
(S,(N /* s1),(||.s1.|| ")) . N2 is Element of the carrier of S
(N /* s1) . N2 is Element of the carrier of S
(||.s1.|| ") . N2 is V11() real ext-real Element of REAL
((||.s1.|| ") . N2) * ((N /* s1) . N2) is Element of the carrier of S
the Mult of S is V16() V19(K7(REAL, the carrier of S)) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of S), the carrier of S))
K7(REAL, the carrier of S) is set
K7(K7(REAL, the carrier of S), the carrier of S) is set
K6(K7(K7(REAL, the carrier of S), the carrier of S)) is set
K249( the Mult of S,((||.s1.|| ") . N2),((N /* s1) . N2)) is set
((||.s1.|| ") . N2) * (N /. (s1 . N2)) is Element of the carrier of S
K249( the Mult of S,((||.s1.|| ") . N2),(N /. (s1 . N2))) is set
(S,(N /* s1),(||.s1.|| ")) . 0 is Element of the carrier of S
lim (S,(N /* s1),(||.s1.|| ")) is Element of the carrier of S
x0 /\ (dom f) is Element of K6( the carrier of T)
N2 is Element of the carrier of S
L is Element of the carrier of T
f /. L is Element of the carrier of S
f . L is set
R is Element of BoundedLinearOperators (T,S)
L is Element of the carrier of T
R is Neighbourhood of L
f /. L is Element of the carrier of S
s1 is V16() V19( the carrier of T) V20( the carrier of S) Function-like (T,S) Element of K6(K7( the carrier of T, the carrier of S))
x is Element of the carrier of T
f /. x is Element of the carrier of S
(f /. x) - (f /. L) is Element of the carrier of S
- (f /. L) is Element of the carrier of S
K192(S,(f /. x),(- (f /. L))) is Element of the carrier of S
x - L is Element of the carrier of T
- L is Element of the carrier of T
K192(T,x,(- L)) is Element of the carrier of T
R . (x - L) is Element of the carrier of S
s1 /. (x - L) is Element of the carrier of S
(R . (x - L)) + (s1 /. (x - L)) is Element of the carrier of S
s1 . (x - L) is set
N2 - (f /. L) is Element of the carrier of S
K192(S,N2,(- (f /. L))) is Element of the carrier of S
N2 - N2 is Element of the carrier of S
- N2 is Element of the carrier of S
K192(S,N2,(- N2)) is Element of the carrier of S
(0. S) + (0. S) is Element of the carrier of S
L is Element of the carrier of T
(T,S,f,x0) /. L is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
R is Neighbourhood of L
f /. L is Element of the carrier of S
x is Element of the carrier of T
f /. x is Element of the carrier of S
(f /. x) - (f /. L) is Element of the carrier of S
- (f /. L) is Element of the carrier of S
K192(S,(f /. x),(- (f /. L))) is Element of the carrier of S
x - L is Element of the carrier of T
- L is Element of the carrier of T
K192(T,x,(- L)) is Element of the carrier of T
R . (x - L) is Element of the carrier of S
s1 /. (x - L) is Element of the carrier of S
(R . (x - L)) + (s1 /. (x - L)) is Element of the carrier of S
s1 . (x - L) is set
N2 - (f /. L) is Element of the carrier of S
K192(S,N2,(- (f /. L))) is Element of the carrier of S
N2 - N2 is Element of the carrier of S
- N2 is Element of the carrier of S
K192(S,N2,(- N2)) is Element of the carrier of S
(0. S) + (0. S) is Element of the carrier of S
(T,S,f,L) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,S))
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
K7( the carrier of T, the carrier of T) is set
K6(K7( the carrier of T, the carrier of T)) is set
K6( the carrier of T) is set
R_NormSpace_of_BoundedLinearOperators (T,T) is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators (T,T)) is V1() set
FuncUnit T is Element of BoundedLinearOperators (T,T)
R_VectorSpace_of_LinearOperators (T,T) is non empty V87() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators (T,T)) is V1() set
BoundedLinearOperators (T,T) is V1() Element of K6( the carrier of (R_VectorSpace_of_LinearOperators (T,T)))
K6( the carrier of (R_VectorSpace_of_LinearOperators (T,T))) is set
S is V16() V19( the carrier of T) V20( the carrier of T) Function-like Element of K6(K7( the carrier of T, the carrier of T))
dom S is Element of K6( the carrier of T)
f is V11() real ext-real Element of REAL
f * (FuncUnit T) is Element of BoundedLinearOperators (T,T)
x0 is Element of the carrier of T
N is Element of K6( the carrier of T)
(T,T,S,N) is V16() V19( the carrier of T) V20( the carrier of (R_NormSpace_of_BoundedLinearOperators (T,T))) Function-like Element of K6(K7( the carrier of T, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,T))))
K7( the carrier of T, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,T))) is set
K6(K7( the carrier of T, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,T)))) is set
Zero_ ((BoundedLinearOperators (T,T)),(R_VectorSpace_of_LinearOperators (T,T))) is Element of BoundedLinearOperators (T,T)
Add_ ((BoundedLinearOperators (T,T)),(R_VectorSpace_of_LinearOperators (T,T))) is V16() V19(K7((BoundedLinearOperators (T,T)),(BoundedLinearOperators (T,T)))) V20( BoundedLinearOperators (T,T)) Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators (T,T)),(BoundedLinearOperators (T,T))),(BoundedLinearOperators (T,T))))
K7((BoundedLinearOperators (T,T)),(BoundedLinearOperators (T,T))) is set
K7(K7((BoundedLinearOperators (T,T)),(BoundedLinearOperators (T,T))),(BoundedLinearOperators (T,T))) is set
K6(K7(K7((BoundedLinearOperators (T,T)),(BoundedLinearOperators (T,T))),(BoundedLinearOperators (T,T)))) is set
Mult_ ((BoundedLinearOperators (T,T)),(R_VectorSpace_of_LinearOperators (T,T))) is V16() V19(K7(REAL,(BoundedLinearOperators (T,T)))) V20( BoundedLinearOperators (T,T)) Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators (T,T))),(BoundedLinearOperators (T,T))))
K7(REAL,(BoundedLinearOperators (T,T))) is set
K7(K7(REAL,(BoundedLinearOperators (T,T))),(BoundedLinearOperators (T,T))) is set
K6(K7(K7(REAL,(BoundedLinearOperators (T,T))),(BoundedLinearOperators (T,T)))) is set
BoundedLinearOperatorsNorm (T,T) is V1() V16() V19( BoundedLinearOperators (T,T)) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators (T,T)),REAL))
K7((BoundedLinearOperators (T,T)),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((BoundedLinearOperators (T,T)),REAL)) is set
NORMSTR(# (BoundedLinearOperators (T,T)),(Zero_ ((BoundedLinearOperators (T,T)),(R_VectorSpace_of_LinearOperators (T,T)))),(Add_ ((BoundedLinearOperators (T,T)),(R_VectorSpace_of_LinearOperators (T,T)))),(Mult_ ((BoundedLinearOperators (T,T)),(R_VectorSpace_of_LinearOperators (T,T)))),(BoundedLinearOperatorsNorm (T,T)) #) is strict NORMSTR
L is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,T))
f * L is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,T))
the Mult of (R_NormSpace_of_BoundedLinearOperators (T,T)) is V16() V19(K7(REAL, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,T)))) V20( the carrier of (R_NormSpace_of_BoundedLinearOperators (T,T))) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,T))), the carrier of (R_NormSpace_of_BoundedLinearOperators (T,T))))
K7(REAL, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,T))) is set
K7(K7(REAL, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,T))), the carrier of (R_NormSpace_of_BoundedLinearOperators (T,T))) is set
K6(K7(K7(REAL, the carrier of (R_NormSpace_of_BoundedLinearOperators (T,T))), the carrier of (R_NormSpace_of_BoundedLinearOperators (T,T)))) is set
K249( the Mult of (R_NormSpace_of_BoundedLinearOperators (T,T)),f,L) is set
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
the carrier of T --> (0. T) is V1() V16() V19( the carrier of T) V20( the carrier of T) Function-like constant total quasi_total Element of K6(K7( the carrier of T, the carrier of T))
s1 is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,T))
N2 is V16() V19( the carrier of T) V20( the carrier of T) Function-like Element of K6(K7( the carrier of T, the carrier of T))
dom N2 is Element of K6( the carrier of T)
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
L is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total non-zero (T, 0. T) Element of K6(K7(NAT, the carrier of T))
R is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
L . R is Element of the carrier of T
N2 /. (L . R) is Element of the carrier of T
N2 . (L . R) is set
rng L is Element of K6( the carrier of T)
N2 /* L is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
||.L.|| is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
||.L.|| " is V1() V16() V19( NAT ) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(T,(N2 /* L),(||.L.|| ")) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(T,(N2 /* L),(||.L.|| ")) . R is Element of the carrier of T
(N2 /* L) . R is Element of the carrier of T
(||.L.|| ") . R is V11() real ext-real Element of REAL
((||.L.|| ") . R) * ((N2 /* L) . R) is Element of the carrier of T
the Mult of T is V16() V19(K7(REAL, the carrier of T)) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of T), the carrier of T))
K7(REAL, the carrier of T) is set
K7(K7(REAL, the carrier of T), the carrier of T) is set
K6(K7(K7(REAL, the carrier of T), the carrier of T)) is set
K249( the Mult of T,((||.L.|| ") . R),((N2 /* L) . R)) is set
((||.L.|| ") . R) * (N2 /. (L . R)) is Element of the carrier of T
K249( the Mult of T,((||.L.|| ") . R),(N2 /. (L . R))) is set
(T,(N2 /* L),(||.L.|| ")) . 0 is Element of the carrier of T
lim (T,(N2 /* L),(||.L.|| ")) is Element of the carrier of T
R is Element of the carrier of T
x is Neighbourhood of R
S /. R is Element of the carrier of T
L is V16() V19( the carrier of T) V20( the carrier of T) Function-like (T,T) Element of K6(K7( the carrier of T, the carrier of T))
y is Element of the carrier of T
S /. y is Element of the carrier of T
(S /. y) - (S /. R) is Element of the carrier of T
- (S /. R) is Element of the carrier of T
K192(T,(S /. y),(- (S /. R))) is Element of the carrier of T
y - R is Element of the carrier of T
- R is Element of the carrier of T
K192(T,y,(- R)) is Element of the carrier of T
s1 . (y - R) is Element of the carrier of T
L /. (y - R) is Element of the carrier of T
(s1 . (y - R)) + (L /. (y - R)) is Element of the carrier of T
L . (y - R) is set
id the carrier of T is V1() V16() V19( the carrier of T) V20( the carrier of T) Function-like one-to-one total quasi_total Element of K6(K7( the carrier of T, the carrier of T))
(id the carrier of T) . (y - R) is Element of the carrier of T
f * (y - R) is Element of the carrier of T
K249( the Mult of T,f,(y - R)) is set
(FuncUnit T) . (y - R) is Element of the carrier of T
f * ((FuncUnit T) . (y - R)) is Element of the carrier of T
K249( the Mult of T,f,((FuncUnit T) . (y - R))) is set
f * y is Element of the carrier of T
K249( the Mult of T,f,y) is set
(f * y) + x0 is Element of the carrier of T
((f * y) + x0) - (S /. R) is Element of the carrier of T
K192(T,((f * y) + x0),(- (S /. R))) is Element of the carrier of T
f * R is Element of the carrier of T
K249( the Mult of T,f,R) is set
(f * R) + x0 is Element of the carrier of T
((f * y) + x0) - ((f * R) + x0) is Element of the carrier of T
- ((f * R) + x0) is Element of the carrier of T
K192(T,((f * y) + x0),(- ((f * R) + x0))) is Element of the carrier of T
x0 - ((f * R) + x0) is Element of the carrier of T
K192(T,x0,(- ((f * R) + x0))) is Element of the carrier of T
(f * y) + (x0 - ((f * R) + x0)) is Element of the carrier of T
x0 - (f * R) is Element of the carrier of T
- (f * R) is Element of the carrier of T
K192(T,x0,(- (f * R))) is Element of the carrier of T
(x0 - (f * R)) - x0 is Element of the carrier of T
- x0 is Element of the carrier of T
K192(T,(x0 - (f * R)),(- x0)) is Element of the carrier of T
(f * y) + ((x0 - (f * R)) - x0) is Element of the carrier of T
x0 + (- (f * R)) is Element of the carrier of T
(x0 + (- (f * R))) - x0 is Element of the carrier of T
K192(T,(x0 + (- (f * R))),(- x0)) is Element of the carrier of T
(f * y) + ((x0 + (- (f * R))) - x0) is Element of the carrier of T
x0 - x0 is Element of the carrier of T
K192(T,x0,(- x0)) is Element of the carrier of T
(- (f * R)) + (x0 - x0) is Element of the carrier of T
(f * y) + ((- (f * R)) + (x0 - x0)) is Element of the carrier of T
(- (f * R)) + (0. T) is Element of the carrier of T
(f * y) + ((- (f * R)) + (0. T)) is Element of the carrier of T
(f * y) - (f * R) is Element of the carrier of T
K192(T,(f * y),(- (f * R))) is Element of the carrier of T
R is Element of the carrier of T
(T,T,S,N) /. R is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,T))
x is Neighbourhood of R
S /. R is Element of the carrier of T
y is Element of the carrier of T
S /. y is Element of the carrier of T
(S /. y) - (S /. R) is Element of the carrier of T
- (S /. R) is Element of the carrier of T
K192(T,(S /. y),(- (S /. R))) is Element of the carrier of T
y - R is Element of the carrier of T
- R is Element of the carrier of T
K192(T,y,(- R)) is Element of the carrier of T
s1 . (y - R) is Element of the carrier of T
L /. (y - R) is Element of the carrier of T
(s1 . (y - R)) + (L /. (y - R)) is Element of the carrier of T
L . (y - R) is set
id the carrier of T is V1() V16() V19( the carrier of T) V20( the carrier of T) Function-like one-to-one total quasi_total Element of K6(K7( the carrier of T, the carrier of T))
(id the carrier of T) . (y - R) is Element of the carrier of T
f * (y - R) is Element of the carrier of T
K249( the Mult of T,f,(y - R)) is set
(FuncUnit T) . (y - R) is Element of the carrier of T
f * ((FuncUnit T) . (y - R)) is Element of the carrier of T
K249( the Mult of T,f,((FuncUnit T) . (y - R))) is set
f * y is Element of the carrier of T
K249( the Mult of T,f,y) is set
(f * y) + x0 is Element of the carrier of T
((f * y) + x0) - (S /. R) is Element of the carrier of T
K192(T,((f * y) + x0),(- (S /. R))) is Element of the carrier of T
f * R is Element of the carrier of T
K249( the Mult of T,f,R) is set
(f * R) + x0 is Element of the carrier of T
((f * y) + x0) - ((f * R) + x0) is Element of the carrier of T
- ((f * R) + x0) is Element of the carrier of T
K192(T,((f * y) + x0),(- ((f * R) + x0))) is Element of the carrier of T
x0 - ((f * R) + x0) is Element of the carrier of T
K192(T,x0,(- ((f * R) + x0))) is Element of the carrier of T
(f * y) + (x0 - ((f * R) + x0)) is Element of the carrier of T
x0 - (f * R) is Element of the carrier of T
- (f * R) is Element of the carrier of T
K192(T,x0,(- (f * R))) is Element of the carrier of T
(x0 - (f * R)) - x0 is Element of the carrier of T
- x0 is Element of the carrier of T
K192(T,(x0 - (f * R)),(- x0)) is Element of the carrier of T
(f * y) + ((x0 - (f * R)) - x0) is Element of the carrier of T
x0 + (- (f * R)) is Element of the carrier of T
(x0 + (- (f * R))) - x0 is Element of the carrier of T
K192(T,(x0 + (- (f * R))),(- x0)) is Element of the carrier of T
(f * y) + ((x0 + (- (f * R))) - x0) is Element of the carrier of T
x0 - x0 is Element of the carrier of T
K192(T,x0,(- x0)) is Element of the carrier of T
(- (f * R)) + (x0 - x0) is Element of the carrier of T
(f * y) + ((- (f * R)) + (x0 - x0)) is Element of the carrier of T
(- (f * R)) + (0. T) is Element of the carrier of T
(f * y) + ((- (f * R)) + (0. T)) is Element of the carrier of T
(f * y) - (f * R) is Element of the carrier of T
K192(T,(f * y),(- (f * R))) is Element of the carrier of T
(T,T,S,R) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (T,T))
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
K7( the carrier of S, the carrier of T) is set
K6(K7( the carrier of S, the carrier of T)) is set
f is V16() V19( the carrier of S) V20( the carrier of T) Function-like Element of K6(K7( the carrier of S, the carrier of T))
x0 is Element of the carrier of S
dom f is Element of K6( the carrier of S)
K6( the carrier of S) is set
R_NormSpace_of_BoundedLinearOperators (S,T) is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T)) is V1() set
f /. x0 is Element of the carrier of T
N is Neighbourhood of x0
L is V11() real ext-real Element of REAL
{ b₁ where b₁ is Element of the carrier of S : not L <= ||.(b₁ - x0).|| } is set
K7(NAT, the carrier of S) is set
K6(K7(NAT, the carrier of S)) is set
NAT --> x0 is V1() V16() V19( NAT ) V20( the carrier of S) Function-like constant total quasi_total Element of K6(K7(NAT, the carrier of S))
s1 is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
rng s1 is Element of K6( the carrier of S)
lim s1 is Element of the carrier of S
N2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
R is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
L is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
0. S is V68(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
R is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
L . R is Element of the carrier of S
s1 . R is Element of the carrier of S
R . R is Element of the carrier of S
(s1 . R) - (R . R) is Element of the carrier of S
- (R . R) is Element of the carrier of S
K192(S,(s1 . R),(- (R . R))) is Element of the carrier of S
(s1 . R) - x0 is Element of the carrier of S
- x0 is Element of the carrier of S
K192(S,(s1 . R),(- x0)) is Element of the carrier of S
L ^\ N2 is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total subsequence of L
R is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
(L ^\ N2) . R is Element of the carrier of S
R + N2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
L . (R + N2) is Element of the carrier of S
R ^\ N2 is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total subsequence of R
s1 - R is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
lim R is Element of the carrier of S
R . 0 is Element of the carrier of S
lim L is Element of the carrier of S
x0 - x0 is Element of the carrier of S
K192(S,x0,(- x0)) is Element of the carrier of S
lim (L ^\ N2) is Element of the carrier of S
x is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total non-zero (S, 0. S) Element of K6(K7(NAT, the carrier of S))
R is V1() V16() V19( NAT ) V20( the carrier of S) Function-like constant total quasi_total Element of K6(K7(NAT, the carrier of S))
x + R is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
f /* (x + R) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
f /* R is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(f /* (x + R)) - (f /* R) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
((f /* (x + R)) - (f /* R)) + (f /* R) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
(((f /* (x + R)) - (f /* R)) + (f /* R)) . y is Element of the carrier of T
((f /* (x + R)) - (f /* R)) . y is Element of the carrier of T
(f /* R) . y is Element of the carrier of T
(((f /* (x + R)) - (f /* R)) . y) + ((f /* R) . y) is Element of the carrier of T
(f /* (x + R)) . y is Element of the carrier of T
((f /* (x + R)) . y) - ((f /* R) . y) is Element of the carrier of T
- ((f /* R) . y) is Element of the carrier of T
K192(T,((f /* (x + R)) . y),(- ((f /* R) . y))) is Element of the carrier of T
(((f /* (x + R)) . y) - ((f /* R) . y)) + ((f /* R) . y) is Element of the carrier of T
((f /* R) . y) - ((f /* R) . y) is Element of the carrier of T
K192(T,((f /* R) . y),(- ((f /* R) . y))) is Element of the carrier of T
((f /* (x + R)) . y) - (((f /* R) . y) - ((f /* R) . y)) is Element of the carrier of T
- (((f /* R) . y) - ((f /* R) . y)) is Element of the carrier of T
K192(T,((f /* (x + R)) . y),(- (((f /* R) . y) - ((f /* R) . y)))) is Element of the carrier of T
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
((f /* (x + R)) . y) - (0. T) is Element of the carrier of T
- (0. T) is Element of the carrier of T
K192(T,((f /* (x + R)) . y),(- (0. T))) is Element of the carrier of T
y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
(x + R) . y is Element of the carrier of S
(s1 - R) + R is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
((s1 - R) + R) ^\ N2 is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total subsequence of (s1 - R) + R
(((s1 - R) + R) ^\ N2) . y is Element of the carrier of S
y + N2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
((s1 - R) + R) . (y + N2) is Element of the carrier of S
(s1 - R) . (y + N2) is Element of the carrier of S
R . (y + N2) is Element of the carrier of S
((s1 - R) . (y + N2)) + (R . (y + N2)) is Element of the carrier of S
s1 . (y + N2) is Element of the carrier of S
(s1 . (y + N2)) - (R . (y + N2)) is Element of the carrier of S
- (R . (y + N2)) is Element of the carrier of S
K192(S,(s1 . (y + N2)),(- (R . (y + N2)))) is Element of the carrier of S
((s1 . (y + N2)) - (R . (y + N2))) + (R . (y + N2)) is Element of the carrier of S
(R . (y + N2)) - (R . (y + N2)) is Element of the carrier of S
K192(S,(R . (y + N2)),(- (R . (y + N2)))) is Element of the carrier of S
(s1 . (y + N2)) - ((R . (y + N2)) - (R . (y + N2))) is Element of the carrier of S
- ((R . (y + N2)) - (R . (y + N2))) is Element of the carrier of S
K192(S,(s1 . (y + N2)),(- ((R . (y + N2)) - (R . (y + N2))))) is Element of the carrier of S
(s1 . (y + N2)) - (0. S) is Element of the carrier of S
- (0. S) is Element of the carrier of S
K192(S,(s1 . (y + N2)),(- (0. S))) is Element of the carrier of S
N2 + y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
s1 . (N2 + y) is Element of the carrier of S
s1 ^\ N2 is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total subsequence of s1
(s1 ^\ N2) . y is Element of the carrier of S
f /* (s1 ^\ N2) is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
f /* s1 is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
(f /* s1) ^\ N2 is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total subsequence of f /* s1
rng R is trivial Element of K6( the carrier of S)
{x0} is set
y is set
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
(R ^\ N2) . n is Element of the carrier of S
n + N2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
R . (n + N2) is Element of the carrier of S
y is set
R . 0 is Element of the carrier of S
0 + N2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
R . (0 + N2) is Element of the carrier of S
y is V11() real ext-real Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
y1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
(f /* R) . y1 is Element of the carrier of T
((f /* R) . y1) - (f /. x0) is Element of the carrier of T
- (f /. x0) is Element of the carrier of T
K192(T,((f /* R) . y1),(- (f /. x0))) is Element of the carrier of T
||.(((f /* R) . y1) - (f /. x0)).|| is V11() real ext-real Element of REAL
R . y1 is Element of the carrier of S
f /. (R . y1) is Element of the carrier of T
(f /. (R . y1)) - (f /. x0) is Element of the carrier of T
K192(T,(f /. (R . y1)),(- (f /. x0))) is Element of the carrier of T
||.((f /. (R . y1)) - (f /. x0)).|| is V11() real ext-real Element of REAL
y1 + N2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
R . (y1 + N2) is Element of the carrier of S
f /. (R . (y1 + N2)) is Element of the carrier of T
(f /. (R . (y1 + N2))) - (f /. x0) is Element of the carrier of T
K192(T,(f /. (R . (y1 + N2))),(- (f /. x0))) is Element of the carrier of T
||.((f /. (R . (y1 + N2))) - (f /. x0)).|| is V11() real ext-real Element of REAL
(f /. x0) - (f /. x0) is Element of the carrier of T
K192(T,(f /. x0),(- (f /. x0))) is Element of the carrier of T
||.((f /. x0) - (f /. x0)).|| is V11() real ext-real Element of REAL
||.(0. T).|| is V11() real ext-real Element of REAL
rng (x + R) is Element of K6( the carrier of S)
y is set
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
(x + R) . n is Element of the carrier of S
(((s1 - R) + R) ^\ N2) . n is Element of the carrier of S
n + N2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
((s1 - R) + R) . (n + N2) is Element of the carrier of S
(s1 - R) . (n + N2) is Element of the carrier of S
R . (n + N2) is Element of the carrier of S
((s1 - R) . (n + N2)) + (R . (n + N2)) is Element of the carrier of S
s1 . (n + N2) is Element of the carrier of S
(s1 . (n + N2)) - (R . (n + N2)) is Element of the carrier of S
- (R . (n + N2)) is Element of the carrier of S
K192(S,(s1 . (n + N2)),(- (R . (n + N2)))) is Element of the carrier of S
((s1 . (n + N2)) - (R . (n + N2))) + (R . (n + N2)) is Element of the carrier of S
(R . (n + N2)) - (R . (n + N2)) is Element of the carrier of S
K192(S,(R . (n + N2)),(- (R . (n + N2)))) is Element of the carrier of S
(s1 . (n + N2)) - ((R . (n + N2)) - (R . (n + N2))) is Element of the carrier of S
- ((R . (n + N2)) - (R . (n + N2))) is Element of the carrier of S
K192(S,(s1 . (n + N2)),(- ((R . (n + N2)) - (R . (n + N2))))) is Element of the carrier of S
(s1 . (n + N2)) - (0. S) is Element of the carrier of S
K192(S,(s1 . (n + N2)),(- (0. S))) is Element of the carrier of S
N2 + n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V129() V169() V170() V171() V172() V173() V174() V175() Element of NAT
s1 . (N2 + n) is Element of the carrier of S
((x + R) . n) - x0 is Element of the carrier of S
K192(S,((x + R) . n),(- x0)) is Element of the carrier of S
||.(((x + R) . n) - x0).|| is V11() real ext-real Element of REAL
y1 is Element of the carrier of S
lim ((f /* (x + R)) - (f /* R)) is Element of the carrier of T
lim (f /* R) is Element of the carrier of T
lim (((f /* (x + R)) - (f /* R)) + (f /* R)) is Element of the carrier of T
(0. T) + (f /. x0) is Element of the carrier of T
lim (f /* s1) is Element of the carrier of T
T is set
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
f is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of f is V1() non trivial set
K7( the carrier of S, the carrier of f) is set
K6(K7( the carrier of S, the carrier of f)) is set
x0 is V16() V19( the carrier of S) V20( the carrier of f) Function-like Element of K6(K7( the carrier of S, the carrier of f))
dom x0 is Element of K6( the carrier of S)
K6( the carrier of S) is set
x0 | T is V16() V19( the carrier of S) V20( the carrier of f) Function-like Element of K6(K7( the carrier of S, the carrier of f))
N is Element of the carrier of S
T is set
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
f is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of f is V1() non trivial set
K7( the carrier of f, the carrier of S) is set
K6(K7( the carrier of f, the carrier of S)) is set
K6( the carrier of f) is set
x0 is V16() V19( the carrier of f) V20( the carrier of S) Function-like Element of K6(K7( the carrier of f, the carrier of S))
N is Element of K6( the carrier of f)
dom x0 is Element of K6( the carrier of f)
x0 | N is V16() V19( the carrier of f) V20( the carrier of S) Function-like Element of K6(K7( the carrier of f, the carrier of S))
L is Element of the carrier of f
R is Neighbourhood of L
x0 | T is V16() V19( the carrier of f) V20( the carrier of S) Function-like Element of K6(K7( the carrier of f, the carrier of S))
dom (x0 | T) is Element of K6( the carrier of f)
R_NormSpace_of_BoundedLinearOperators (f,S) is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators (f,S)) is V1() set
(x0 | T) /. L is Element of the carrier of S
s1 is Neighbourhood of L
N2 is Neighbourhood of L
dom (x0 | N) is Element of K6( the carrier of f)
(x0 | N) /. L is Element of the carrier of S
(dom x0) /\ T is Element of K6( the carrier of f)
(dom x0) /\ N is Element of K6( the carrier of f)
L is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (f,S))
R is V16() V19( the carrier of f) V20( the carrier of S) Function-like (f,S) Element of K6(K7( the carrier of f, the carrier of S))
L is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (f,S))
R is V16() V19( the carrier of f) V20( the carrier of S) Function-like (f,S) Element of K6(K7( the carrier of f, the carrier of S))
x is Element of the carrier of f
(x0 | N) /. x is Element of the carrier of S
((x0 | N) /. x) - ((x0 | N) /. L) is Element of the carrier of S
- ((x0 | N) /. L) is Element of the carrier of S
K192(S,((x0 | N) /. x),(- ((x0 | N) /. L))) is Element of the carrier of S
x - L is Element of the carrier of f
- L is Element of the carrier of f
K192(f,x,(- L)) is Element of the carrier of f
L . (x - L) is Element of the carrier of S
R /. (x - L) is Element of the carrier of S
(L . (x - L)) + (R /. (x - L)) is Element of the carrier of S
(x0 | T) /. x is Element of the carrier of S
((x0 | T) /. x) - ((x0 | T) /. L) is Element of the carrier of S
- ((x0 | T) /. L) is Element of the carrier of S
K192(S,((x0 | T) /. x),(- ((x0 | T) /. L))) is Element of the carrier of S
x0 /. L is Element of the carrier of S
((x0 | T) /. x) - (x0 /. L) is Element of the carrier of S
- (x0 /. L) is Element of the carrier of S
K192(S,((x0 | T) /. x),(- (x0 /. L))) is Element of the carrier of S
x0 /. x is Element of the carrier of S
(x0 /. x) - (x0 /. L) is Element of the carrier of S
K192(S,(x0 /. x),(- (x0 /. L))) is Element of the carrier of S
(x0 /. x) - ((x0 | N) /. L) is Element of the carrier of S
K192(S,(x0 /. x),(- ((x0 | N) /. L))) is Element of the carrier of S
T is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of T is V1() non trivial set
0. T is V68(T) Element of the carrier of T
the ZeroF of T is Element of the carrier of T
S is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of S is V1() non trivial set
K7( the carrier of S, the carrier of T) is set
K6(K7( the carrier of S, the carrier of T)) is set
0. S is V68(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
f is V16() V19( the carrier of S) V20( the carrier of T) Function-like Element of K6(K7( the carrier of S, the carrier of T))
dom f is Element of K6( the carrier of S)
K6( the carrier of S) is set
x0 is Element of the carrier of S
f /. x0 is Element of the carrier of T
(S,T,f,x0) is V16() Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T))
R_NormSpace_of_BoundedLinearOperators (S,T) is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators (S,T)) is V1() set
N is Neighbourhood of x0
BoundedLinearOperators (S,T) is V1() Element of K6( the carrier of (R_VectorSpace_of_LinearOperators (S,T)))
R_VectorSpace_of_LinearOperators (S,T) is non empty V87() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators (S,T)) is V1() set
K6( the carrier of (R_VectorSpace_of_LinearOperators (S,T))) is set
Zero_ ((BoundedLinearOperators (S,T)),(R_VectorSpace_of_LinearOperators (S,T))) is Element of BoundedLinearOperators (S,T)
Add_ ((BoundedLinearOperators (S,T)),(R_VectorSpace_of_LinearOperators (S,T))) is V16() V19(K7((BoundedLinearOperators (S,T)),(BoundedLinearOperators (S,T)))) V20( BoundedLinearOperators (S,T)) Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators (S,T)),(BoundedLinearOperators (S,T))),(BoundedLinearOperators (S,T))))
K7((BoundedLinearOperators (S,T)),(BoundedLinearOperators (S,T))) is set
K7(K7((BoundedLinearOperators (S,T)),(BoundedLinearOperators (S,T))),(BoundedLinearOperators (S,T))) is set
K6(K7(K7((BoundedLinearOperators (S,T)),(BoundedLinearOperators (S,T))),(BoundedLinearOperators (S,T)))) is set
Mult_ ((BoundedLinearOperators (S,T)),(R_VectorSpace_of_LinearOperators (S,T))) is V16() V19(K7(REAL,(BoundedLinearOperators (S,T)))) V20( BoundedLinearOperators (S,T)) Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators (S,T))),(BoundedLinearOperators (S,T))))
K7(REAL,(BoundedLinearOperators (S,T))) is set
K7(K7(REAL,(BoundedLinearOperators (S,T))),(BoundedLinearOperators (S,T))) is set
K6(K7(K7(REAL,(BoundedLinearOperators (S,T))),(BoundedLinearOperators (S,T)))) is set
BoundedLinearOperatorsNorm (S,T) is V1() V16() V19( BoundedLinearOperators (S,T)) V20( REAL ) Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators (S,T)),REAL))
K7((BoundedLinearOperators (S,T)),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((BoundedLinearOperators (S,T)),REAL)) is set
NORMSTR(# (BoundedLinearOperators (S,T)),(Zero_ ((BoundedLinearOperators (S,T)),(R_VectorSpace_of_LinearOperators (S,T)))),(Add_ ((BoundedLinearOperators (S,T)),(R_VectorSpace_of_LinearOperators (S,T)))),(Mult_ ((BoundedLinearOperators (S,T)),(R_VectorSpace_of_LinearOperators (S,T)))),(BoundedLinearOperatorsNorm (S,T)) #) is strict NORMSTR
R is V16() V19( the carrier of S) V20( the carrier of T) Function-like (S,T) Element of K6(K7( the carrier of S, the carrier of T))
R is V16() V19( the carrier of S) V20( the carrier of T) Function-like (S,T) Element of K6(K7( the carrier of S, the carrier of T))
R /. (0. S) is Element of the carrier of T
(f /. x0) - (f /. x0) is Element of the carrier of T
- (f /. x0) is Element of the carrier of T
K192(T,(f /. x0),(- (f /. x0))) is Element of the carrier of T
L is Element of BoundedLinearOperators (S,T)
x0 - x0 is Element of the carrier of S
- x0 is Element of the carrier of S
K192(S,x0,(- x0)) is Element of the carrier of S
L . (x0 - x0) is Element of the carrier of T
R /. (x0 - x0) is Element of the carrier of T
(L . (x0 - x0)) + (R /. (x0 - x0)) is Element of the carrier of T
L . (0. S) is Element of the carrier of T
(L . (0. S)) + (R /. (x0 - x0)) is Element of the carrier of T
(L . (0. S)) + (R /. (0. S)) is Element of the carrier of T
modetrans (L,S,T) is V1() V16() V19( the carrier of S) V20( the carrier of T) Function-like total quasi_total V155(S,T) V156(S,T) Lipschitzian Element of K6(K7( the carrier of S, the carrier of T))
(modetrans (L,S,T)) . (0. S) is Element of the carrier of T
0 * (0. S) is Element of the carrier of S
the Mult of S is V16() V19(K7(REAL, the carrier of S)) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of S), the carrier of S))
K7(REAL, the carrier of S) is set
K7(K7(REAL, the carrier of S), the carrier of S) is set
K6(K7(K7(REAL, the carrier of S), the carrier of S)) is set
K249( the Mult of S,0,(0. S)) is set
(modetrans (L,S,T)) . (0 * (0. S)) is Element of the carrier of T
0 * ((modetrans (L,S,T)) . (0. S)) is Element of the carrier of T
the Mult of T is V16() V19(K7(REAL, the carrier of T)) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of T), the carrier of T))
K7(REAL, the carrier of T) is set
K7(K7(REAL, the carrier of T), the carrier of T) is set
K6(K7(K7(REAL, the carrier of T), the carrier of T)) is set
K249( the Mult of T,0,((modetrans (L,S,T)) . (0. S))) is set
K7(NAT, the carrier of S) is set
K6(K7(NAT, the carrier of S)) is set
s1 is V1() V16() V19( NAT ) V20( the carrier of S) Function-like total quasi_total Element of K6(K7(NAT, the carrier of S))
rng s1 is Element of K6( the carrier of S)
dom R is Element of K6( the carrier of S)
lim s1 is Element of the carrier of S
R /* s1 is V1() V16() V19( NAT ) V20( the carrier of T) Function-like total quasi_total Element of K6(K7(NAT, the carrier of T))
K7(NAT, the carrier of T) is set
K6(K7(NAT, the carrier of T)) is set
lim (R /* s1) is Element of the carrier of T
s1 is Element of the carrier of S
f /. s1 is Element of the carrier of T
(f /. s1) - (f /. x0) is Element of the carrier of T
K192(T,(f /. s1),(- (f /. x0))) is Element of the carrier of T
s1 - x0 is Element of the carrier of S
K192(S,s1,(- x0)) is Element of the carrier of S
(S,T,f,x0) . (s1 - x0) is Element of the carrier of T
R /. (s1 - x0) is Element of the carrier of T
((S,T,f,x0) . (s1 - x0)) + (R /. (s1 - x0)) is Element of the carrier of T
L is V16() V19( the carrier of S) V20( the carrier of T) Function-like (S,T) Element of K6(K7( the carrier of S, the carrier of T))
L /. (0. S) is Element of the carrier of T