REAL is non empty V37() V159() V160() V161() V165() set
NAT is ordinal V37() cardinal limit_cardinal V159() V160() V161() V162() V163() V164() V165() Element of bool REAL
bool REAL is V37() set
COMPLEX is non empty V37() V159() V165() set
RAT is non empty V37() V159() V160() V161() V162() V165() set
INT is non empty V37() V159() V160() V161() V162() V163() V165() set
[:COMPLEX,COMPLEX:] is V37() complex-valued set
bool [:COMPLEX,COMPLEX:] is V37() set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is V37() complex-valued set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is V37() set
[:REAL,REAL:] is V37() complex-valued ext-real-valued real-valued set
bool [:REAL,REAL:] is V37() set
[:[:REAL,REAL:],REAL:] is V37() complex-valued ext-real-valued real-valued set
bool [:[:REAL,REAL:],REAL:] is V37() set
[:RAT,RAT:] is RAT -valued V37() complex-valued ext-real-valued real-valued set
bool [:RAT,RAT:] is V37() set
[:[:RAT,RAT:],RAT:] is RAT -valued V37() complex-valued ext-real-valued real-valued set
bool [:[:RAT,RAT:],RAT:] is V37() set
[:INT,INT:] is RAT -valued INT -valued V37() complex-valued ext-real-valued real-valued set
bool [:INT,INT:] is V37() set
[:[:INT,INT:],INT:] is RAT -valued INT -valued V37() complex-valued ext-real-valued real-valued set
bool [:[:INT,INT:],INT:] is V37() set
[:NAT,NAT:] is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
[:[:NAT,NAT:],NAT:] is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:[:NAT,NAT:],NAT:] is set
omega is ordinal V37() cardinal limit_cardinal V159() V160() V161() V162() V163() V164() V165() set
bool omega is V37() set
bool NAT is V37() set
1 is non empty ordinal natural complex real ext-real positive non negative V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT
[:1,1:] is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:1,1:] is set
[:[:1,1:],1:] is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:[:1,1:],1:] is set
[:[:1,1:],REAL:] is complex-valued ext-real-valued real-valued set
bool [:[:1,1:],REAL:] is set
2 is non empty ordinal natural complex real ext-real positive non negative V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT
[:2,2:] is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
[:[:2,2:],REAL:] is complex-valued ext-real-valued real-valued set
bool [:[:2,2:],REAL:] is set
K428() is non empty strict multMagma
the carrier of K428() is non empty set
<REAL,+> is non empty strict V130() V131() V132() V134() left-invertible right-invertible invertible left-cancelable right-cancelable V186() multMagma
addreal is Relation-like [:REAL,REAL:] -defined [:REAL,REAL:] -defined REAL -valued REAL -valued Function-like quasi_total quasi_total V76( REAL ) V77( REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
multMagma(# REAL,addreal #) is strict multMagma
K434() is non empty strict V132() V134() left-cancelable right-cancelable V186() SubStr of <REAL,+>
<NAT,+> is non empty strict V130() V132() V134() left-cancelable right-cancelable V186() uniquely-decomposable SubStr of K434()
<REAL,*> is non empty strict V130() V132() V134() multMagma
multreal is Relation-like [:REAL,REAL:] -defined [:REAL,REAL:] -defined REAL -valued REAL -valued Function-like quasi_total quasi_total V76( REAL ) V77( REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
multMagma(# REAL,multreal #) is strict multMagma
<NAT,*> is non empty strict V130() V132() V134() uniquely-decomposable SubStr of <REAL,*>
K522() is V207() TopStruct
the carrier of K522() is V159() V160() V161() set
RealSpace is non empty strict Reflexive discerning V86() triangle Discerning V207() MetrStruct
K527() is non empty strict TopSpace-like V207() TopStruct
{} is empty ordinal Function-like functional cardinal {} -element FinSequence-membered V159() V160() V161() V162() V163() V164() V165() set
0 is empty ordinal natural complex real ext-real non positive non negative Function-like functional V33() V34() V37() cardinal {} -element FinSequence-membered V159() V160() V161() V162() V163() V164() V165() Element of NAT
3 is non empty ordinal natural complex real ext-real positive non negative V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT
Seg 1 is non empty trivial V37() 1 -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= 1 ) } is set
{1} is non empty trivial 1 -element V159() V160() V161() V162() V163() V164() set
Seg 2 is non empty V37() 2 -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= 2 ) } is set
{1,2} is non empty V159() V160() V161() V162() V163() V164() set
sqrt 0 is complex real ext-real Element of REAL
sqrt 1 is complex real ext-real Element of REAL
- 1 is complex real ext-real non positive Element of REAL
f is ordinal natural complex real ext-real V37() cardinal set
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
REAL * is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
f is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
f is complex real ext-real set
f . f is complex real ext-real Element of REAL
abs f is complex real ext-real Element of REAL
f is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
f is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
x is complex real ext-real Element of REAL
f . x is complex real ext-real Element of REAL
abs x is complex real ext-real Element of REAL
f . x is complex real ext-real Element of REAL
() is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.f.| is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
rng |.f.| is V159() V160() V161() Element of bool REAL
f (#) () is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
dom () is set
rng f is V159() V160() V161() Element of bool REAL
dom |.f.| is V159() V160() V161() V162() V163() V164() Element of bool NAT
dom f is V159() V160() V161() V162() V163() V164() Element of bool NAT
dom (f (#) ()) is V159() V160() V161() V162() V163() V164() Element of bool NAT
x is set
|.f.| . x is complex real ext-real Element of REAL
f . x is complex real ext-real Element of REAL
abs (f . x) is complex real ext-real Element of REAL
() . (f . x) is complex real ext-real Element of REAL
(f (#) ()) . x is complex real ext-real Element of REAL
x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
M is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f is ordinal natural complex real ext-real V37() cardinal set
f |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
Seg f is V37() f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= f ) } is set
(Seg f) --> 0 is Relation-like Seg f -defined RAT -valued INT -valued {0} -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of bool [:(Seg f),{0}:]
{0} is non empty trivial 1 -element V159() V160() V161() V162() V163() V164() set
[:(Seg f),{0}:] is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg f),{0}:] is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
Seg f is V37() f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= f ) } is set
(Seg f) --> 0 is Relation-like Seg f -defined RAT -valued INT -valued {0} -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of bool [:(Seg f),{0}:]
[:(Seg f),{0}:] is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg f),{0}:] is set
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
f is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
- f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
compreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
f (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
- 1 is complex real ext-real non positive set
(- 1) (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K83(REAL) is non empty Relation-like REAL -defined REAL -valued Function-like one-to-one V26( REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] ((- 1),K83(REAL)) is set
f (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT
x -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = x } is set
M is Relation-like NAT -defined REAL -valued Function-like V37() x -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of x -tuples_on REAL
- M is Relation-like NAT -defined REAL -valued Function-like V37() x -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of x -tuples_on REAL
M (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
f is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
x is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
f + x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (f,x) is set
M is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT
M -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = M } is set
M is Relation-like NAT -defined REAL -valued Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of M -tuples_on REAL
s is Relation-like NAT -defined REAL -valued Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of M -tuples_on REAL
M + s is Relation-like NAT -defined REAL -valued Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of M -tuples_on REAL
addreal .: (M,s) is set
f - x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
diffreal is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
addreal * (K83(REAL),compreal) is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
diffreal .: (f,x) is set
- x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (f,(- x)) is set
M is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT
M -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = M } is set
M is Relation-like NAT -defined REAL -valued Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of M -tuples_on REAL
s is Relation-like NAT -defined REAL -valued Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of M -tuples_on REAL
M - s is Relation-like NAT -defined REAL -valued Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of M -tuples_on REAL
diffreal .: (M,s) is set
- s is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
s (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) s is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
s (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M + (- s) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (M,(- s)) is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
f is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
x is complex real ext-real set
x (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] (x,K83(REAL)) is set
f (#) (x multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT
M -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = M } is set
M is Relation-like NAT -defined REAL -valued Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of M -tuples_on REAL
x * M is Relation-like NAT -defined REAL -valued Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of M -tuples_on REAL
M (#) (x multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
f is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
|.f.| is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) () is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
f is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
f ^2 is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
sqrreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
f (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: (f,f) is set
f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
sqr f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: (f,f) is set
Sum (sqr f) is complex real ext-real Element of REAL
sqrt (Sum (sqr f)) is complex real ext-real Element of REAL
f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
len f is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT
((len f)) is non empty functional FinSequence-membered FinSequenceSet of REAL
(len f) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = len f } is set
rng f is V159() V160() V161() Element of bool REAL
f is ordinal natural complex real ext-real V37() cardinal set
(f) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
f |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
Seg f is V37() f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= f ) } is set
(Seg f) --> 0 is Relation-like Seg f -defined RAT -valued INT -valued {0} -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of bool [:(Seg f),{0}:]
[:(Seg f),{0}:] is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg f),{0}:] is set
(f,(f)) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
(f) (#) () is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT
() . 0 is complex real ext-real Element of REAL
f |-> (() . 0) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
Seg f is V37() f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= f ) } is set
(Seg f) --> (() . 0) is Relation-like Seg f -defined {(() . 0)} -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:(Seg f),{(() . 0)}:]
{(() . 0)} is non empty trivial 1 -element V159() V160() V161() set
[:(Seg f),{(() . 0)}:] is complex-valued ext-real-valued real-valued set
bool [:(Seg f),{(() . 0)}:] is set
abs 0 is complex real ext-real V34() Element of REAL
f |-> (abs 0) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
(Seg f) --> (abs 0) is Relation-like Seg f -defined RAT -valued {(abs 0)} -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:(Seg f),{(abs 0)}:]
{(abs 0)} is non empty trivial 1 -element V159() V160() V161() V162() set
[:(Seg f),{(abs 0)}:] is RAT -valued complex-valued ext-real-valued real-valued set
bool [:(Seg f),{(abs 0)}:] is set
f is Relation-like Function-like complex-valued set
- f is Relation-like Function-like complex-valued set
(- 1) (#) f is Relation-like Function-like complex-valued set
|.(- f).| is Relation-like Function-like complex-valued ext-real-valued real-valued set
|.f.| is Relation-like Function-like complex-valued ext-real-valued real-valued set
dom |.(- f).| is set
dom (- f) is set
dom |.f.| is set
dom f is set
f is set
|.(- f).| . f is complex real ext-real Element of REAL
(- f) . f is complex set
abs ((- f) . f) is complex real ext-real Element of REAL
f . f is complex set
- (f . f) is complex Element of COMPLEX
abs (- (f . f)) is complex real ext-real Element of REAL
abs (f . f) is complex real ext-real Element of REAL
|.f.| . f is complex real ext-real Element of REAL
f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f is complex real ext-real set
f * f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] (f,K83(REAL)) is set
f (#) (f multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((f * f)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f * f) (#) () is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f (#) () is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
abs f is complex real ext-real Element of REAL
(abs f) * (f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(abs f) multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] ((abs f),K83(REAL)) is set
(f) (#) ((abs f) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
f |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
Seg f is V37() f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= f ) } is set
(Seg f) --> 0 is Relation-like Seg f -defined RAT -valued INT -valued {0} -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of bool [:(Seg f),{0}:]
[:(Seg f),{0}:] is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg f),{0}:] is set
((f)) is complex real ext-real Element of REAL
sqr (f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f) (#) (f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f),(f)) is set
Sum (sqr (f)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f))) is complex real ext-real Element of REAL
0 ^2 is complex real ext-real Element of REAL
0 * 0 is empty ordinal complex real ext-real non positive non negative Function-like functional cardinal {} -element FinSequence-membered V159() V160() V161() V162() V163() V164() V165() set
f |-> (0 ^2) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
(Seg f) --> (0 ^2) is Relation-like Seg f -defined {(0 ^2)} -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:(Seg f),{(0 ^2)}:]
{(0 ^2)} is non empty trivial 1 -element V159() V160() V161() set
[:(Seg f),{(0 ^2)}:] is complex-valued ext-real-valued real-valued set
bool [:(Seg f),{(0 ^2)}:] is set
Sum (f |-> (0 ^2)) is complex real ext-real Element of REAL
sqrt (Sum (f |-> (0 ^2))) is complex real ext-real Element of REAL
f * 0 is complex real ext-real Element of REAL
sqrt (f * 0) is complex real ext-real Element of REAL
f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f (#) () is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
sqr (f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f) (#) (f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f),(f)) is set
sqr f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: (f,f) is set
len f is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT
((len f)) is non empty functional FinSequence-membered FinSequenceSet of REAL
(len f) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = len f } is set
M is ordinal natural complex real ext-real V37() cardinal set
Seg (len f) is V37() len f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= len f ) } is set
x is Relation-like NAT -defined REAL -valued Function-like V37() len f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of ((len f))
((len f),x) is Relation-like NAT -defined REAL -valued Function-like V37() len f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (len f) -tuples_on REAL
x (#) () is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
sqr ((len f),x) is Relation-like NAT -defined REAL -valued Function-like V37() len f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (len f) -tuples_on REAL
((len f),x) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((len f),x) (#) ((len f),x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: (((len f),x),((len f),x)) is set
(sqr ((len f),x)) . M is complex real ext-real Element of REAL
((len f),x) . M is complex real ext-real Element of REAL
(((len f),x) . M) ^2 is complex real ext-real Element of REAL
(((len f),x) . M) * (((len f),x) . M) is complex real ext-real set
x . M is complex real ext-real Element of REAL
abs (x . M) is complex real ext-real Element of REAL
(abs (x . M)) ^2 is complex real ext-real Element of REAL
(abs (x . M)) * (abs (x . M)) is complex real ext-real set
(x . M) ^2 is complex real ext-real Element of REAL
(x . M) * (x . M) is complex real ext-real set
((len f),x) is Relation-like NAT -defined REAL -valued Function-like V37() len f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (len f) -tuples_on REAL
x (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: (x,x) is set
((len f),x) . M is complex real ext-real Element of REAL
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
(f) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
f |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
Seg f is V37() f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= f ) } is set
(Seg f) --> 0 is Relation-like Seg f -defined RAT -valued INT -valued {0} -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of bool [:(Seg f),{0}:]
[:(Seg f),{0}:] is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg f),{0}:] is set
f is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
(f) is complex real ext-real Element of REAL
sqr f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: (f,f) is set
Sum (sqr f) is complex real ext-real Element of REAL
sqrt (Sum (sqr f)) is complex real ext-real Element of REAL
x is ordinal natural complex real ext-real V37() cardinal set
f . x is complex real ext-real Element of REAL
(f,f) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
Sum (f,f) is complex real ext-real Element of REAL
(f,f) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
f (#) () is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
sqr (f,f) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
(f,f) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,f) (#) (f,f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f,f),(f,f)) is set
Sum (sqr (f,f)) is complex real ext-real Element of REAL
(f,f) . x is complex real ext-real Element of REAL
(f |-> 0) . x is empty ordinal complex real ext-real non positive non negative Function-like functional cardinal {} -element FinSequence-membered V159() V160() V161() V162() V163() V164() V165() Element of REAL
M is complex real ext-real Element of REAL
abs M is complex real ext-real Element of REAL
f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f) is complex real ext-real Element of REAL
sqr f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: (f,f) is set
Sum (sqr f) is complex real ext-real Element of REAL
sqrt (Sum (sqr f)) is complex real ext-real Element of REAL
f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f) is complex real ext-real Element of REAL
sqr f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: (f,f) is set
Sum (sqr f) is complex real ext-real Element of REAL
sqrt (Sum (sqr f)) is complex real ext-real Element of REAL
f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
- f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((- f)) is complex real ext-real non negative Element of REAL
sqr (- f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(- f) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- f) (#) (- f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((- f),(- f)) is set
Sum (sqr (- f)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (- f))) is complex real ext-real Element of REAL
(f) is complex real ext-real non negative Element of REAL
sqr f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: (f,f) is set
Sum (sqr f) is complex real ext-real Element of REAL
sqrt (Sum (sqr f)) is complex real ext-real Element of REAL
((- f)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(- f) (#) () is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
sqr ((- f)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((- f)) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((- f)) (#) ((- f)) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: (((- f)),((- f))) is set
Sum (sqr ((- f))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((- f)))) is complex real ext-real Element of REAL
(f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f (#) () is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
sqr (f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f) (#) (f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f),(f)) is set
Sum (sqr (f)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f))) is complex real ext-real Element of REAL
f is complex real ext-real set
abs f is complex real ext-real Element of REAL
f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f * f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] (f,K83(REAL)) is set
f (#) (f multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((f * f)) is complex real ext-real non negative Element of REAL
sqr (f * f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f * f) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f * f) (#) (f * f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f * f),(f * f)) is set
Sum (sqr (f * f)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f * f))) is complex real ext-real Element of REAL
(f) is complex real ext-real non negative Element of REAL
sqr f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: (f,f) is set
Sum (sqr f) is complex real ext-real Element of REAL
sqrt (Sum (sqr f)) is complex real ext-real Element of REAL
(abs f) * (f) is complex real ext-real Element of REAL
len f is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT
((len f)) is non empty functional FinSequence-membered FinSequenceSet of REAL
(len f) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = len f } is set
(abs f) ^2 is complex real ext-real Element of REAL
(abs f) * (abs f) is complex real ext-real set
M is Relation-like NAT -defined REAL -valued Function-like V37() len f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of ((len f))
((len f),M) is Relation-like NAT -defined REAL -valued Function-like V37() len f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (len f) -tuples_on REAL
M (#) () is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
sqr ((len f),M) is Relation-like NAT -defined REAL -valued Function-like V37() len f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (len f) -tuples_on REAL
((len f),M) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((len f),M) (#) ((len f),M) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: (((len f),M),((len f),M)) is set
Sum (sqr ((len f),M)) is complex real ext-real Element of REAL
((len f),M,f) is Relation-like NAT -defined REAL -valued Function-like V37() len f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of ((len f))
M (#) (f multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((len f),((len f),M,f)) is Relation-like NAT -defined REAL -valued Function-like V37() len f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (len f) -tuples_on REAL
((len f),M,f) (#) () is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
sqr ((len f),((len f),M,f)) is Relation-like NAT -defined REAL -valued Function-like V37() len f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (len f) -tuples_on REAL
((len f),((len f),M,f)) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((len f),((len f),M,f)) (#) ((len f),((len f),M,f)) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: (((len f),((len f),M,f)),((len f),((len f),M,f))) is set
Sum (sqr ((len f),((len f),M,f))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((len f),((len f),M,f)))) is complex real ext-real Element of REAL
(abs f) * ((len f),M) is Relation-like NAT -defined REAL -valued Function-like V37() len f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (len f) -tuples_on REAL
(abs f) multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] ((abs f),K83(REAL)) is set
((len f),M) (#) ((abs f) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
sqr ((abs f) * ((len f),M)) is Relation-like NAT -defined REAL -valued Function-like V37() len f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (len f) -tuples_on REAL
((abs f) * ((len f),M)) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((abs f) * ((len f),M)) (#) ((abs f) * ((len f),M)) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: (((abs f) * ((len f),M)),((abs f) * ((len f),M))) is set
Sum (sqr ((abs f) * ((len f),M))) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((abs f) * ((len f),M)))) is complex real ext-real Element of REAL
((abs f) ^2) * (sqr ((len f),M)) is Relation-like NAT -defined REAL -valued Function-like V37() len f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (len f) -tuples_on REAL
((abs f) ^2) multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] (((abs f) ^2),K83(REAL)) is set
(sqr ((len f),M)) (#) (((abs f) ^2) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum (((abs f) ^2) * (sqr ((len f),M))) is complex real ext-real Element of REAL
sqrt (Sum (((abs f) ^2) * (sqr ((len f),M)))) is complex real ext-real Element of REAL
((abs f) ^2) * (Sum (sqr ((len f),M))) is complex real ext-real Element of REAL
sqrt (((abs f) ^2) * (Sum (sqr ((len f),M)))) is complex real ext-real Element of REAL
sqrt ((abs f) ^2) is complex real ext-real Element of REAL
sqrt (Sum (sqr ((len f),M))) is complex real ext-real Element of REAL
(sqrt ((abs f) ^2)) * (sqrt (Sum (sqr ((len f),M)))) is complex real ext-real Element of REAL
(abs f) * (sqrt (Sum (sqr ((len f),M)))) is complex real ext-real Element of REAL
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
f is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
(f) is complex real ext-real non negative Element of REAL
sqr f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: (f,f) is set
Sum (sqr f) is complex real ext-real Element of REAL
sqrt (Sum (sqr f)) is complex real ext-real Element of REAL
x is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
(f,f,x) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
addreal .: (f,x) is set
((f,f,x)) is complex real ext-real non negative Element of REAL
sqr (f,f,x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f,f,x) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,f,x) (#) (f,f,x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f,f,x),(f,f,x)) is set
Sum (sqr (f,f,x)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f,f,x))) is complex real ext-real Element of REAL
(x) is complex real ext-real non negative Element of REAL
sqr x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
x (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: (x,x) is set
Sum (sqr x) is complex real ext-real Element of REAL
sqrt (Sum (sqr x)) is complex real ext-real Element of REAL
(f) + (x) is complex real ext-real non negative Element of REAL
(f,(f,f,x)) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
Sum (f,(f,f,x)) is complex real ext-real Element of REAL
(f,f) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
f (#) () is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
sqr (f,f) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
(f,f) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,f) (#) (f,f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f,f),(f,f)) is set
Sum (sqr (f,f)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f,f))) is complex real ext-real Element of REAL
Seg f is V37() f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= f ) } is set
(f,(f,f,x)) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
(f,f,x) (#) () is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
sqr (f,(f,f,x)) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
(f,(f,f,x)) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,(f,f,x)) (#) (f,(f,f,x)) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f,(f,f,x)),(f,(f,f,x))) is set
(f,x) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
x (#) () is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,f) + (f,x) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
addreal .: ((f,f),(f,x)) is set
sqr ((f,f) + (f,x)) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
((f,f) + (f,x)) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((f,f) + (f,x)) (#) ((f,f) + (f,x)) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: (((f,f) + (f,x)),((f,f) + (f,x))) is set
M is ordinal natural complex real ext-real V37() cardinal set
(sqr (f,(f,f,x))) . M is complex real ext-real Element of REAL
(sqr ((f,f) + (f,x))) . M is complex real ext-real Element of REAL
len (f,f,x) is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT
dom (f,f,x) is f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
(f,f) . M is complex real ext-real Element of REAL
(f,x) . M is complex real ext-real Element of REAL
(f,f,x) . M is complex real ext-real Element of REAL
f . M is complex real ext-real Element of REAL
x . M is complex real ext-real Element of REAL
y is complex real ext-real Element of REAL
r22 is complex real ext-real Element of REAL
y + r22 is complex real ext-real Element of REAL
abs (y + r22) is complex real ext-real Element of REAL
abs y is complex real ext-real Element of REAL
abs r22 is complex real ext-real Element of REAL
(abs y) + (abs r22) is complex real ext-real Element of REAL
f is complex real ext-real Element of REAL
abs f is complex real ext-real Element of REAL
s is complex real ext-real Element of REAL
(abs y) + s is complex real ext-real Element of REAL
M is complex real ext-real Element of REAL
M + s is complex real ext-real Element of REAL
(f,(f,f,x)) . M is complex real ext-real Element of REAL
((f,f) + (f,x)) . M is complex real ext-real Element of REAL
abs912 is complex real ext-real Element of REAL
len ((f,f) + (f,x)) is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT
dom ((f,f) + (f,x)) is f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
abs12 is complex real ext-real Element of REAL
abs912 ^2 is complex real ext-real Element of REAL
abs912 * abs912 is complex real ext-real set
abs12 ^2 is complex real ext-real Element of REAL
abs12 * abs12 is complex real ext-real set
mlt ((f,f),(f,x)) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
multreal .: ((f,f),(f,x)) is set
Sum (mlt ((f,f),(f,x))) is complex real ext-real Element of REAL
(Sum (mlt ((f,f),(f,x)))) ^2 is complex real ext-real Element of REAL
(Sum (mlt ((f,f),(f,x)))) * (Sum (mlt ((f,f),(f,x)))) is complex real ext-real set
sqrt ((Sum (mlt ((f,f),(f,x)))) ^2) is complex real ext-real Element of REAL
sqr (f,x) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
(f,x) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,x) (#) (f,x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f,x),(f,x)) is set
Sum (sqr (f,x)) is complex real ext-real Element of REAL
(Sum (sqr (f,f))) * (Sum (sqr (f,x))) is complex real ext-real Element of REAL
sqrt ((Sum (sqr (f,f))) * (Sum (sqr (f,x)))) is complex real ext-real Element of REAL
M is ordinal natural complex real ext-real V37() cardinal set
(mlt ((f,f),(f,x))) . M is complex real ext-real Element of REAL
f . M is complex real ext-real Element of REAL
x . M is complex real ext-real Element of REAL
(f,f) . M is complex real ext-real Element of REAL
s is complex real ext-real Element of REAL
abs s is complex real ext-real Element of REAL
(f,x) . M is complex real ext-real Element of REAL
f is complex real ext-real Element of REAL
abs f is complex real ext-real Element of REAL
(abs s) * (abs f) is complex real ext-real Element of REAL
len (mlt ((f,f),(f,x))) is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT
dom (mlt ((f,f),(f,x))) is f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
2 * (Sum (mlt ((f,f),(f,x)))) is complex real ext-real Element of REAL
2 * (sqrt ((Sum (sqr (f,f))) * (Sum (sqr (f,x))))) is complex real ext-real Element of REAL
(Sum (sqr (f,f))) + (2 * (Sum (mlt ((f,f),(f,x))))) is complex real ext-real Element of REAL
(Sum (sqr (f,f))) + (2 * (sqrt ((Sum (sqr (f,f))) * (Sum (sqr (f,x)))))) is complex real ext-real Element of REAL
((Sum (sqr (f,f))) + (2 * (Sum (mlt ((f,f),(f,x)))))) + (Sum (sqr (f,x))) is complex real ext-real Element of REAL
((Sum (sqr (f,f))) + (2 * (sqrt ((Sum (sqr (f,f))) * (Sum (sqr (f,x))))))) + (Sum (sqr (f,x))) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f,x))) is complex real ext-real Element of REAL
Sum (sqr ((f,f) + (f,x))) is complex real ext-real Element of REAL
2 * (mlt ((f,f),(f,x))) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
2 multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] (2,K83(REAL)) is set
(mlt ((f,f),(f,x))) (#) (2 multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(sqr (f,f)) + (2 * (mlt ((f,f),(f,x)))) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
addreal .: ((sqr (f,f)),(2 * (mlt ((f,f),(f,x))))) is set
((sqr (f,f)) + (2 * (mlt ((f,f),(f,x))))) + (sqr (f,x)) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
addreal .: (((sqr (f,f)) + (2 * (mlt ((f,f),(f,x))))),(sqr (f,x))) is set
Sum (((sqr (f,f)) + (2 * (mlt ((f,f),(f,x))))) + (sqr (f,x))) is complex real ext-real Element of REAL
Sum ((sqr (f,f)) + (2 * (mlt ((f,f),(f,x))))) is complex real ext-real Element of REAL
(Sum ((sqr (f,f)) + (2 * (mlt ((f,f),(f,x)))))) + (Sum (sqr (f,x))) is complex real ext-real Element of REAL
Sum (2 * (mlt ((f,f),(f,x)))) is complex real ext-real Element of REAL
(Sum (sqr (f,f))) + (Sum (2 * (mlt ((f,f),(f,x))))) is complex real ext-real Element of REAL
((Sum (sqr (f,f))) + (Sum (2 * (mlt ((f,f),(f,x)))))) + (Sum (sqr (f,x))) is complex real ext-real Element of REAL
Sum (sqr (f,(f,f,x))) is complex real ext-real Element of REAL
(sqrt (Sum (sqr (f,f)))) * (sqrt (Sum (sqr (f,x)))) is complex real ext-real Element of REAL
2 * ((sqrt (Sum (sqr (f,f)))) * (sqrt (Sum (sqr (f,x))))) is complex real ext-real Element of REAL
(Sum (sqr (f,f))) + (2 * ((sqrt (Sum (sqr (f,f)))) * (sqrt (Sum (sqr (f,x)))))) is complex real ext-real Element of REAL
((Sum (sqr (f,f))) + (2 * ((sqrt (Sum (sqr (f,f)))) * (sqrt (Sum (sqr (f,x))))))) + (Sum (sqr (f,x))) is complex real ext-real Element of REAL
(sqrt (Sum (sqr (f,x)))) ^2 is complex real ext-real Element of REAL
(sqrt (Sum (sqr (f,x)))) * (sqrt (Sum (sqr (f,x)))) is complex real ext-real set
(sqrt (Sum (sqr (f,f)))) ^2 is complex real ext-real Element of REAL
(sqrt (Sum (sqr (f,f)))) * (sqrt (Sum (sqr (f,f)))) is complex real ext-real set
sqrt (Sum (f,(f,f,x))) is complex real ext-real Element of REAL
(sqrt (Sum (sqr (f,f)))) + (sqrt (Sum (sqr (f,x)))) is complex real ext-real Element of REAL
((sqrt (Sum (sqr (f,f)))) + (sqrt (Sum (sqr (f,x))))) ^2 is complex real ext-real Element of REAL
((sqrt (Sum (sqr (f,f)))) + (sqrt (Sum (sqr (f,x))))) * ((sqrt (Sum (sqr (f,f)))) + (sqrt (Sum (sqr (f,x))))) is complex real ext-real set
sqrt (((sqrt (Sum (sqr (f,f)))) + (sqrt (Sum (sqr (f,x))))) ^2) is complex real ext-real Element of REAL
(f,x) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
Sum (f,x) is complex real ext-real Element of REAL
sqrt (Sum (f,x)) is complex real ext-real Element of REAL
(sqrt (Sum (sqr (f,f)))) + (sqrt (Sum (f,x))) is complex real ext-real Element of REAL
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
f is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
(f) is complex real ext-real non negative Element of REAL
sqr f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: (f,f) is set
Sum (sqr f) is complex real ext-real Element of REAL
sqrt (Sum (sqr f)) is complex real ext-real Element of REAL
x is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
(f,f,x) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
diffreal .: (f,x) is set
- x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (f,(- x)) is set
((f,f,x)) is complex real ext-real non negative Element of REAL
sqr (f,f,x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f,f,x) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,f,x) (#) (f,f,x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f,f,x),(f,f,x)) is set
Sum (sqr (f,f,x)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f,f,x))) is complex real ext-real Element of REAL
(x) is complex real ext-real non negative Element of REAL
sqr x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
x (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: (x,x) is set
Sum (sqr x) is complex real ext-real Element of REAL
sqrt (Sum (sqr x)) is complex real ext-real Element of REAL
(f) + (x) is complex real ext-real non negative Element of REAL
(f,x) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
((f,x)) is complex real ext-real non negative Element of REAL
sqr (f,x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f,x) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,x) (#) (f,x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f,x),(f,x)) is set
Sum (sqr (f,x)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f,x))) is complex real ext-real Element of REAL
(f) + ((f,x)) is complex real ext-real non negative Element of REAL
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
f is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
(f) is complex real ext-real non negative Element of REAL
sqr f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: (f,f) is set
Sum (sqr f) is complex real ext-real Element of REAL
sqrt (Sum (sqr f)) is complex real ext-real Element of REAL
x is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
(x) is complex real ext-real non negative Element of REAL
sqr x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
x (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: (x,x) is set
Sum (sqr x) is complex real ext-real Element of REAL
sqrt (Sum (sqr x)) is complex real ext-real Element of REAL
(f) - (x) is complex real ext-real Element of REAL
(f,f,x) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
addreal .: (f,x) is set
((f,f,x)) is complex real ext-real non negative Element of REAL
sqr (f,f,x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f,f,x) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,f,x) (#) (f,f,x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f,f,x),(f,f,x)) is set
Sum (sqr (f,f,x)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f,f,x))) is complex real ext-real Element of REAL
M is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
M is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
M + M is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
addreal .: (M,M) is set
(M + M) - M is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
diffreal .: ((M + M),M) is set
- M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(M + M) + (- M) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: ((M + M),(- M)) is set
((f,f,x)) + (x) is complex real ext-real non negative Element of REAL
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
f is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
(f) is complex real ext-real non negative Element of REAL
sqr f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: (f,f) is set
Sum (sqr f) is complex real ext-real Element of REAL
sqrt (Sum (sqr f)) is complex real ext-real Element of REAL
x is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
(x) is complex real ext-real non negative Element of REAL
sqr x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
x (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: (x,x) is set
Sum (sqr x) is complex real ext-real Element of REAL
sqrt (Sum (sqr x)) is complex real ext-real Element of REAL
(f) - (x) is complex real ext-real Element of REAL
(f,f,x) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
diffreal .: (f,x) is set
- x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (f,(- x)) is set
((f,f,x)) is complex real ext-real non negative Element of REAL
sqr (f,f,x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f,f,x) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,f,x) (#) (f,f,x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f,f,x),(f,f,x)) is set
Sum (sqr (f,f,x)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f,f,x))) is complex real ext-real Element of REAL
M is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
M is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
M - M is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
diffreal .: (M,M) is set
- M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M + (- M) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (M,(- M)) is set
(M - M) + M is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
addreal .: ((M - M),M) is set
((f,f,x)) + (x) is complex real ext-real non negative Element of REAL
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
f is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
x is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
(f,f,x) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
diffreal .: (f,x) is set
- x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (f,(- x)) is set
((f,f,x)) is complex real ext-real non negative Element of REAL
sqr (f,f,x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f,f,x) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,f,x) (#) (f,f,x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f,f,x),(f,f,x)) is set
Sum (sqr (f,f,x)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f,f,x))) is complex real ext-real Element of REAL
M is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
M is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
M - M is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
diffreal .: (M,M) is set
- M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M + (- M) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (M,(- M)) is set
(f) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
f |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
Seg f is V37() f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= f ) } is set
(Seg f) --> 0 is Relation-like Seg f -defined RAT -valued INT -valued {0} -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of bool [:(Seg f),{0}:]
[:(Seg f),{0}:] is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg f),{0}:] is set
M is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
M is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
M - M is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
diffreal .: (M,M) is set
- M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M + (- M) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (M,(- M)) is set
(f) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
f |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
Seg f is V37() f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= f ) } is set
(Seg f) --> 0 is Relation-like Seg f -defined RAT -valued INT -valued {0} -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of bool [:(Seg f),{0}:]
[:(Seg f),{0}:] is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg f),{0}:] is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
f is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
(f,f,f) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
diffreal .: (f,f) is set
- f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (f,(- f)) is set
((f,f,f)) is complex real ext-real non negative Element of REAL
sqr (f,f,f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f,f,f) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,f,f) (#) (f,f,f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f,f,f),(f,f,f)) is set
Sum (sqr (f,f,f)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f,f,f))) is complex real ext-real Element of REAL
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
f is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
x is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
(f,f,x) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
diffreal .: (f,x) is set
- x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (f,(- x)) is set
((f,f,x)) is complex real ext-real non negative Element of REAL
sqr (f,f,x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f,f,x) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,f,x) (#) (f,f,x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f,f,x),(f,f,x)) is set
Sum (sqr (f,f,x)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f,f,x))) is complex real ext-real Element of REAL
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
f is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
x is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
(f,f,x) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
diffreal .: (f,x) is set
- x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (f,(- x)) is set
((f,f,x)) is complex real ext-real non negative Element of REAL
sqr (f,f,x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f,f,x) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,f,x) (#) (f,f,x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f,f,x),(f,f,x)) is set
Sum (sqr (f,f,x)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f,f,x))) is complex real ext-real Element of REAL
(f,x,f) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
diffreal .: (x,f) is set
- f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x + (- f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (x,(- f)) is set
((f,x,f)) is complex real ext-real non negative Element of REAL
sqr (f,x,f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f,x,f) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,x,f) (#) (f,x,f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f,x,f),(f,x,f)) is set
Sum (sqr (f,x,f)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f,x,f))) is complex real ext-real Element of REAL
M is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
M is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
M - M is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
diffreal .: (M,M) is set
- M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M + (- M) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (M,(- M)) is set
- (M - M) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
(M - M) (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) (M - M) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(M - M) (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((- (M - M))) is complex real ext-real non negative Element of REAL
sqr (- (M - M)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(- (M - M)) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- (M - M)) (#) (- (M - M)) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((- (M - M)),(- (M - M))) is set
Sum (sqr (- (M - M))) is complex real ext-real Element of REAL
sqrt (Sum (sqr (- (M - M)))) is complex real ext-real Element of REAL
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
f is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
x is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
(f,f,x) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
diffreal .: (f,x) is set
- x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (f,(- x)) is set
((f,f,x)) is complex real ext-real non negative Element of REAL
sqr (f,f,x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f,f,x) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,f,x) (#) (f,f,x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f,f,x),(f,f,x)) is set
Sum (sqr (f,f,x)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f,f,x))) is complex real ext-real Element of REAL
M is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
(f,f,M) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
diffreal .: (f,M) is set
- M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- M) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (f,(- M)) is set
((f,f,M)) is complex real ext-real non negative Element of REAL
sqr (f,f,M) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f,f,M) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,f,M) (#) (f,f,M) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f,f,M),(f,f,M)) is set
Sum (sqr (f,f,M)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f,f,M))) is complex real ext-real Element of REAL
(f,M,x) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
diffreal .: (M,x) is set
M + (- x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (M,(- x)) is set
((f,M,x)) is complex real ext-real non negative Element of REAL
sqr (f,M,x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f,M,x) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,M,x) (#) (f,M,x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f,M,x),(f,M,x)) is set
Sum (sqr (f,M,x)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f,M,x))) is complex real ext-real Element of REAL
((f,f,M)) + ((f,M,x)) is complex real ext-real non negative Element of REAL
M is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
f is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
M - f is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
diffreal .: (M,f) is set
- f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M + (- f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (M,(- f)) is set
(M - f) + f is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
addreal .: ((M - f),f) is set
s is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
((M - f) + f) - s is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
diffreal .: (((M - f) + f),s) is set
- s is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
s (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) s is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
s (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((M - f) + f) + (- s) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (((M - f) + f),(- s)) is set
((((M - f) + f) - s)) is complex real ext-real non negative Element of REAL
sqr (((M - f) + f) - s) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(((M - f) + f) - s) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(((M - f) + f) - s) (#) (((M - f) + f) - s) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((((M - f) + f) - s),(((M - f) + f) - s)) is set
Sum (sqr (((M - f) + f) - s)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (((M - f) + f) - s))) is complex real ext-real Element of REAL
(f,(f,f,M),(f,M,x)) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
addreal .: ((f,f,M),(f,M,x)) is set
((f,(f,f,M),(f,M,x))) is complex real ext-real non negative Element of REAL
sqr (f,(f,f,M),(f,M,x)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f,(f,f,M),(f,M,x)) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,(f,f,M),(f,M,x)) (#) (f,(f,f,M),(f,M,x)) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f,(f,f,M),(f,M,x)),(f,(f,f,M),(f,M,x))) is set
Sum (sqr (f,(f,f,M),(f,M,x))) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f,(f,f,M),(f,M,x)))) is complex real ext-real Element of REAL
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
[:(f),(f):] is set
[:[:(f),(f):],REAL:] is complex-valued ext-real-valued real-valued set
bool [:[:(f),(f):],REAL:] is set
f is Relation-like [:(f),(f):] -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:[:(f),(f):],REAL:]
x is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
M is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
f . (x,M) is complex real ext-real Element of REAL
(f,x,M) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
diffreal .: (x,M) is set
- M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x + (- M) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (x,(- M)) is set
((f,x,M)) is complex real ext-real non negative Element of REAL
sqr (f,x,M) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f,x,M) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,x,M) (#) (f,x,M) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f,x,M),(f,x,M)) is set
Sum (sqr (f,x,M)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f,x,M))) is complex real ext-real Element of REAL
f is Relation-like [:(f),(f):] -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:[:(f),(f):],REAL:]
x is Relation-like [:(f),(f):] -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:[:(f),(f):],REAL:]
M is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT
(M) is non empty functional FinSequence-membered FinSequenceSet of REAL
M -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = M } is set
[:(M),(M):] is set
[:[:(M),(M):],REAL:] is complex-valued ext-real-valued real-valued set
bool [:[:(M),(M):],REAL:] is set
M is Relation-like [:(M),(M):] -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:[:(M),(M):],REAL:]
f is Relation-like NAT -defined REAL -valued Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (M)
y is Relation-like NAT -defined REAL -valued Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (M)
M . (f,y) is complex real ext-real Element of REAL
(M,f,y) is Relation-like NAT -defined REAL -valued Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (M)
diffreal .: (f,y) is set
- y is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
y (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) y is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
y (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- y) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (f,(- y)) is set
((M,f,y)) is complex real ext-real non negative Element of REAL
sqr (M,f,y) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(M,f,y) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(M,f,y) (#) (M,f,y) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((M,f,y),(M,f,y)) is set
Sum (sqr (M,f,y)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (M,f,y))) is complex real ext-real Element of REAL
s is Relation-like [:(M),(M):] -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:[:(M),(M):],REAL:]
s . (f,y) is complex real ext-real Element of REAL
f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f - f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: (f,f) is set
- f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (f,(- f)) is set
sqr (f - f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f - f) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f - f) (#) (f - f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f - f),(f - f)) is set
f - f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: (f,f) is set
- f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (f,(- f)) is set
sqr (f - f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f - f) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f - f) (#) (f - f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f - f),(f - f)) is set
(f - f) ^2 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
[:NAT,REAL:] is V37() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is V37() set
f ^2 is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: (f,f) is set
f (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: (f,f) is set
2 (#) (f (#) f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
2 multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] (2,K83(REAL)) is set
(f (#) f) (#) (2 multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f ^2) - (2 (#) (f (#) f)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: ((f ^2),(2 (#) (f (#) f))) is set
- (2 (#) (f (#) f)) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(2 (#) (f (#) f)) (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) (2 (#) (f (#) f)) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(2 (#) (f (#) f)) (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f ^2) + (- (2 (#) (f (#) f))) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: ((f ^2),(- (2 (#) (f (#) f)))) is set
f ^2 is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: (f,f) is set
((f ^2) - (2 (#) (f (#) f))) + (f ^2) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (((f ^2) - (2 (#) (f (#) f))),(f ^2)) is set
sqr f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mlt (f,f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
2 * (mlt (f,f)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(mlt (f,f)) (#) (2 multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
- (2 * (mlt (f,f))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(2 * (mlt (f,f))) (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) (2 * (mlt (f,f))) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(2 * (mlt (f,f))) (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
sqr f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(- (2 * (mlt (f,f)))) + (sqr f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: ((- (2 * (mlt (f,f)))),(sqr f)) is set
(sqr f) + ((- (2 * (mlt (f,f)))) + (sqr f)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: ((sqr f),((- (2 * (mlt (f,f)))) + (sqr f))) is set
mlt (f,f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
multreal .: (f,f) is set
2 * (mlt (f,f)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(mlt (f,f)) (#) (2 multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(sqr f) - (2 * (mlt (f,f))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: ((sqr f),(2 * (mlt (f,f)))) is set
- (2 * (mlt (f,f))) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(2 * (mlt (f,f))) (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) (2 * (mlt (f,f))) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(2 * (mlt (f,f))) (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(sqr f) + (- (2 * (mlt (f,f)))) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: ((sqr f),(- (2 * (mlt (f,f))))) is set
((sqr f) - (2 * (mlt (f,f)))) + (sqr f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (((sqr f) - (2 * (mlt (f,f)))),(sqr f)) is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
(f) is Relation-like [:(f),(f):] -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:[:(f),(f):],REAL:]
[:(f),(f):] is set
[:[:(f),(f):],REAL:] is complex-valued ext-real-valued real-valued set
bool [:[:(f),(f):],REAL:] is set
f is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
x is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
(f) . (f,x) is complex real ext-real Element of REAL
(f) . (x,f) is complex real ext-real Element of REAL
M is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
(f) . (f,M) is complex real ext-real Element of REAL
(f) . (x,M) is complex real ext-real Element of REAL
((f) . (f,x)) + ((f) . (x,M)) is complex real ext-real Element of REAL
(f) . (x,M) is complex real ext-real Element of REAL
(f,x,M) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
diffreal .: (x,M) is set
- M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x + (- M) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (x,(- M)) is set
((f,x,M)) is complex real ext-real non negative Element of REAL
sqr (f,x,M) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f,x,M) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,x,M) (#) (f,x,M) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f,x,M),(f,x,M)) is set
Sum (sqr (f,x,M)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f,x,M))) is complex real ext-real Element of REAL
(f) . (f,x) is complex real ext-real Element of REAL
(f,f,x) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
diffreal .: (f,x) is set
- x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (f,(- x)) is set
((f,f,x)) is complex real ext-real non negative Element of REAL
sqr (f,f,x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f,f,x) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,f,x) (#) (f,f,x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f,f,x),(f,f,x)) is set
Sum (sqr (f,f,x)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f,f,x))) is complex real ext-real Element of REAL
(f,x,f) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
diffreal .: (x,f) is set
- f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x + (- f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (x,(- f)) is set
((f,x,f)) is complex real ext-real non negative Element of REAL
sqr (f,x,f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f,x,f) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,x,f) (#) (f,x,f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f,x,f),(f,x,f)) is set
Sum (sqr (f,x,f)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f,x,f))) is complex real ext-real Element of REAL
(f) . (x,f) is complex real ext-real Element of REAL
(f) . (f,M) is complex real ext-real Element of REAL
(f,f,M) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
diffreal .: (f,M) is set
f + (- M) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (f,(- M)) is set
((f,f,M)) is complex real ext-real non negative Element of REAL
sqr (f,f,M) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f,f,M) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,f,M) (#) (f,f,M) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f,f,M),(f,f,M)) is set
Sum (sqr (f,f,M)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f,f,M))) is complex real ext-real Element of REAL
((f) . (f,x)) + ((f) . (x,M)) is complex real ext-real Element of REAL
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
(f) is Relation-like [:(f),(f):] -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:[:(f),(f):],REAL:]
[:(f),(f):] is set
[:[:(f),(f):],REAL:] is complex-valued ext-real-valued real-valued set
bool [:[:(f),(f):],REAL:] is set
MetrStruct(# (f),(f) #) is strict MetrStruct
SpaceMetr ((f),(f)) is strict Reflexive discerning V86() triangle MetrStruct
f is ordinal natural complex real ext-real V37() cardinal set
(f) is strict Reflexive discerning V86() triangle MetrStruct
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
(f) is Relation-like [:(f),(f):] -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:[:(f),(f):],REAL:]
[:(f),(f):] is set
[:[:(f),(f):],REAL:] is complex-valued ext-real-valued real-valued set
bool [:[:(f),(f):],REAL:] is set
MetrStruct(# (f),(f) #) is strict MetrStruct
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty strict Reflexive discerning V86() triangle Discerning MetrStruct
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
(f) is Relation-like [:(f),(f):] -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:[:(f),(f):],REAL:]
[:(f),(f):] is set
[:[:(f),(f):],REAL:] is complex-valued ext-real-valued real-valued set
bool [:[:(f),(f):],REAL:] is set
MetrStruct(# (f),(f) #) is strict MetrStruct
TopSpaceMetr (f) is non empty strict TopSpace-like TopStruct
the carrier of (f) is non empty set
Family_open_set (f) is Element of bool (bool the carrier of (f))
bool the carrier of (f) is set
bool (bool the carrier of (f)) is set
TopStruct(# the carrier of (f),(Family_open_set (f)) #) is strict TopStruct
Seg f is V37() f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= f ) } is set
RealVectSpace (Seg f) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
Funcs ((Seg f),REAL) is non empty FUNCTION_DOMAIN of Seg f, REAL
RealFuncZero (Seg f) is Relation-like Seg f -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of Funcs ((Seg f),REAL)
(Seg f) --> 0 is Relation-like Seg f -defined NAT -valued RAT -valued INT -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of bool [:(Seg f),NAT:]
[:(Seg f),NAT:] is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg f),NAT:] is set
RealFuncAdd (Seg f) is Relation-like [:(Funcs ((Seg f),REAL)),(Funcs ((Seg f),REAL)):] -defined Funcs ((Seg f),REAL) -valued Function-like quasi_total Element of bool [:[:(Funcs ((Seg f),REAL)),(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):]
[:(Funcs ((Seg f),REAL)),(Funcs ((Seg f),REAL)):] is set
[:[:(Funcs ((Seg f),REAL)),(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):] is set
bool [:[:(Funcs ((Seg f),REAL)),(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):] is set
RealFuncExtMult (Seg f) is Relation-like [:REAL,(Funcs ((Seg f),REAL)):] -defined Funcs ((Seg f),REAL) -valued Function-like quasi_total Element of bool [:[:REAL,(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):]
[:REAL,(Funcs ((Seg f),REAL)):] is V37() set
[:[:REAL,(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):] is V37() set
bool [:[:REAL,(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):] is V37() set
RLSStruct(# (Funcs ((Seg f),REAL)),(RealFuncZero (Seg f)),(RealFuncAdd (Seg f)),(RealFuncExtMult (Seg f)) #) is strict RLSStruct
the carrier of (RealVectSpace (Seg f)) is non empty set
bool the carrier of (RealVectSpace (Seg f)) is set
bool (bool the carrier of (RealVectSpace (Seg f))) is set
the topology of (TopSpaceMetr (f)) is Element of bool (bool the carrier of (TopSpaceMetr (f)))
the carrier of (TopSpaceMetr (f)) is non empty set
bool the carrier of (TopSpaceMetr (f)) is set
bool (bool the carrier of (TopSpaceMetr (f))) is set
the ZeroF of (RealVectSpace (Seg f)) is Element of the carrier of (RealVectSpace (Seg f))
the addF of (RealVectSpace (Seg f)) is Relation-like [: the carrier of (RealVectSpace (Seg f)), the carrier of (RealVectSpace (Seg f)):] -defined the carrier of (RealVectSpace (Seg f)) -valued Function-like quasi_total Element of bool [:[: the carrier of (RealVectSpace (Seg f)), the carrier of (RealVectSpace (Seg f)):], the carrier of (RealVectSpace (Seg f)):]
[: the carrier of (RealVectSpace (Seg f)), the carrier of (RealVectSpace (Seg f)):] is set
[:[: the carrier of (RealVectSpace (Seg f)), the carrier of (RealVectSpace (Seg f)):], the carrier of (RealVectSpace (Seg f)):] is set
bool [:[: the carrier of (RealVectSpace (Seg f)), the carrier of (RealVectSpace (Seg f)):], the carrier of (RealVectSpace (Seg f)):] is set
the Mult of (RealVectSpace (Seg f)) is Relation-like [:REAL, the carrier of (RealVectSpace (Seg f)):] -defined the carrier of (RealVectSpace (Seg f)) -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of (RealVectSpace (Seg f)):], the carrier of (RealVectSpace (Seg f)):]
[:REAL, the carrier of (RealVectSpace (Seg f)):] is V37() set
[:[:REAL, the carrier of (RealVectSpace (Seg f)):], the carrier of (RealVectSpace (Seg f)):] is V37() set
bool [:[:REAL, the carrier of (RealVectSpace (Seg f)):], the carrier of (RealVectSpace (Seg f)):] is V37() set
M is Element of bool (bool the carrier of (RealVectSpace (Seg f)))
RLTopStruct(# the carrier of (RealVectSpace (Seg f)), the ZeroF of (RealVectSpace (Seg f)), the addF of (RealVectSpace (Seg f)), the Mult of (RealVectSpace (Seg f)),M #) is non empty strict RLTopStruct
M is non empty strict RLTopStruct
the carrier of M is non empty set
the topology of M is Element of bool (bool the carrier of M)
bool the carrier of M is set
bool (bool the carrier of M) is set
TopStruct(# the carrier of M, the topology of M #) is strict TopStruct
the ZeroF of M is Element of the carrier of M
the addF of M is Relation-like [: the carrier of M, the carrier of M:] -defined the carrier of M -valued Function-like quasi_total Element of bool [:[: the carrier of M, the carrier of M:], the carrier of M:]
[: the carrier of M, the carrier of M:] is set
[:[: the carrier of M, the carrier of M:], the carrier of M:] is set
bool [:[: the carrier of M, the carrier of M:], the carrier of M:] is set
the Mult of M is Relation-like [:REAL, the carrier of M:] -defined the carrier of M -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of M:], the carrier of M:]
[:REAL, the carrier of M:] is V37() set
[:[:REAL, the carrier of M:], the carrier of M:] is V37() set
bool [:[:REAL, the carrier of M:], the carrier of M:] is V37() set
RLSStruct(# the carrier of M, the ZeroF of M, the addF of M, the Mult of M #) is strict RLSStruct
f is strict RLTopStruct
the carrier of f is set
the topology of f is Element of bool (bool the carrier of f)
bool the carrier of f is set
bool (bool the carrier of f) is set
TopStruct(# the carrier of f, the topology of f #) is strict TopStruct
the ZeroF of f is Element of the carrier of f
the addF of f is Relation-like [: the carrier of f, the carrier of f:] -defined the carrier of f -valued Function-like quasi_total Element of bool [:[: the carrier of f, the carrier of f:], the carrier of f:]
[: the carrier of f, the carrier of f:] is set
[:[: the carrier of f, the carrier of f:], the carrier of f:] is set
bool [:[: the carrier of f, the carrier of f:], the carrier of f:] is set
the Mult of f is Relation-like [:REAL, the carrier of f:] -defined the carrier of f -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of f:], the carrier of f:]
[:REAL, the carrier of f:] is set
[:[:REAL, the carrier of f:], the carrier of f:] is set
bool [:[:REAL, the carrier of f:], the carrier of f:] is set
RLSStruct(# the carrier of f, the ZeroF of f, the addF of f, the Mult of f #) is strict RLSStruct
x is strict RLTopStruct
the carrier of x is set
the topology of x is Element of bool (bool the carrier of x)
bool the carrier of x is set
bool (bool the carrier of x) is set
TopStruct(# the carrier of x, the topology of x #) is strict TopStruct
the ZeroF of x is Element of the carrier of x
the addF of x is Relation-like [: the carrier of x, the carrier of x:] -defined the carrier of x -valued Function-like quasi_total Element of bool [:[: the carrier of x, the carrier of x:], the carrier of x:]
[: the carrier of x, the carrier of x:] is set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is set
the Mult of x is Relation-like [:REAL, the carrier of x:] -defined the carrier of x -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of x:], the carrier of x:]
[:REAL, the carrier of x:] is set
[:[:REAL, the carrier of x:], the carrier of x:] is set
bool [:[:REAL, the carrier of x:], the carrier of x:] is set
RLSStruct(# the carrier of x, the ZeroF of x, the addF of x, the Mult of x #) is strict RLSStruct
f is ordinal natural complex real ext-real V37() cardinal set
(f) is strict RLTopStruct
the carrier of (f) is set
the topology of (f) is Element of bool (bool the carrier of (f))
bool the carrier of (f) is set
bool (bool the carrier of (f)) is set
TopStruct(# the carrier of (f), the topology of (f) #) is strict TopStruct
(f) is non empty strict Reflexive discerning V86() triangle Discerning MetrStruct
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
(f) is Relation-like [:(f),(f):] -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:[:(f),(f):],REAL:]
[:(f),(f):] is set
[:[:(f),(f):],REAL:] is complex-valued ext-real-valued real-valued set
bool [:[:(f),(f):],REAL:] is set
MetrStruct(# (f),(f) #) is strict MetrStruct
TopSpaceMetr (f) is non empty strict TopSpace-like TopStruct
the carrier of (f) is non empty set
Family_open_set (f) is Element of bool (bool the carrier of (f))
bool the carrier of (f) is set
bool (bool the carrier of (f)) is set
TopStruct(# the carrier of (f),(Family_open_set (f)) #) is strict TopStruct
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty strict RLTopStruct
the carrier of (f) is non empty set
the topology of (f) is Element of bool (bool the carrier of (f))
bool the carrier of (f) is set
bool (bool the carrier of (f)) is set
TopStruct(# the carrier of (f), the topology of (f) #) is strict TopStruct
(f) is non empty strict Reflexive discerning V86() triangle Discerning MetrStruct
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
(f) is Relation-like [:(f),(f):] -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:[:(f),(f):],REAL:]
[:(f),(f):] is set
[:[:(f),(f):],REAL:] is complex-valued ext-real-valued real-valued set
bool [:[:(f),(f):],REAL:] is set
MetrStruct(# (f),(f) #) is strict MetrStruct
TopSpaceMetr (f) is non empty strict TopSpace-like TopStruct
the carrier of (f) is non empty set
Family_open_set (f) is Element of bool (bool the carrier of (f))
bool the carrier of (f) is set
bool (bool the carrier of (f)) is set
TopStruct(# the carrier of (f),(Family_open_set (f)) #) is strict TopStruct
the ZeroF of (f) is Element of the carrier of (f)
the addF of (f) is Relation-like [: the carrier of (f), the carrier of (f):] -defined the carrier of (f) -valued Function-like quasi_total Element of bool [:[: the carrier of (f), the carrier of (f):], the carrier of (f):]
[: the carrier of (f), the carrier of (f):] is set
[:[: the carrier of (f), the carrier of (f):], the carrier of (f):] is set
bool [:[: the carrier of (f), the carrier of (f):], the carrier of (f):] is set
the Mult of (f) is Relation-like [:REAL, the carrier of (f):] -defined the carrier of (f) -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of (f):], the carrier of (f):]
[:REAL, the carrier of (f):] is V37() set
[:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
bool [:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
RLSStruct(# the carrier of (f), the ZeroF of (f), the addF of (f), the Mult of (f) #) is strict RLSStruct
Seg f is V37() f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= f ) } is set
RealVectSpace (Seg f) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
Funcs ((Seg f),REAL) is non empty FUNCTION_DOMAIN of Seg f, REAL
RealFuncZero (Seg f) is Relation-like Seg f -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of Funcs ((Seg f),REAL)
(Seg f) --> 0 is Relation-like Seg f -defined NAT -valued RAT -valued INT -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of bool [:(Seg f),NAT:]
[:(Seg f),NAT:] is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg f),NAT:] is set
RealFuncAdd (Seg f) is Relation-like [:(Funcs ((Seg f),REAL)),(Funcs ((Seg f),REAL)):] -defined Funcs ((Seg f),REAL) -valued Function-like quasi_total Element of bool [:[:(Funcs ((Seg f),REAL)),(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):]
[:(Funcs ((Seg f),REAL)),(Funcs ((Seg f),REAL)):] is set
[:[:(Funcs ((Seg f),REAL)),(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):] is set
bool [:[:(Funcs ((Seg f),REAL)),(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):] is set
RealFuncExtMult (Seg f) is Relation-like [:REAL,(Funcs ((Seg f),REAL)):] -defined Funcs ((Seg f),REAL) -valued Function-like quasi_total Element of bool [:[:REAL,(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):]
[:REAL,(Funcs ((Seg f),REAL)):] is V37() set
[:[:REAL,(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):] is V37() set
bool [:[:REAL,(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):] is V37() set
RLSStruct(# (Funcs ((Seg f),REAL)),(RealFuncZero (Seg f)),(RealFuncAdd (Seg f)),(RealFuncExtMult (Seg f)) #) is strict RLSStruct
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
the topology of (f) is Element of bool (bool the carrier of (f))
bool the carrier of (f) is set
bool (bool the carrier of (f)) is set
TopStruct(# the carrier of (f), the topology of (f) #) is strict TopStruct
(f) is non empty strict Reflexive discerning V86() triangle Discerning MetrStruct
(f) is Relation-like [:(f),(f):] -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:[:(f),(f):],REAL:]
[:(f),(f):] is set
[:[:(f),(f):],REAL:] is complex-valued ext-real-valued real-valued set
bool [:[:(f),(f):],REAL:] is set
MetrStruct(# (f),(f) #) is strict MetrStruct
TopSpaceMetr (f) is non empty strict TopSpace-like TopStruct
the carrier of (f) is non empty set
Family_open_set (f) is Element of bool (bool the carrier of (f))
bool the carrier of (f) is set
bool (bool the carrier of (f)) is set
TopStruct(# the carrier of (f),(Family_open_set (f)) #) is strict TopStruct
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
Seg f is V37() f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= f ) } is set
[:(Seg f),REAL:] is complex-valued ext-real-valued real-valued set
bool [:(Seg f),REAL:] is set
f is Element of the carrier of (f)
the topology of (f) is Element of bool (bool the carrier of (f))
bool the carrier of (f) is set
bool (bool the carrier of (f)) is set
TopStruct(# the carrier of (f), the topology of (f) #) is strict TopStruct
(f) is non empty strict Reflexive discerning V86() triangle Discerning MetrStruct
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
(f) is Relation-like [:(f),(f):] -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:[:(f),(f):],REAL:]
[:(f),(f):] is set
[:[:(f),(f):],REAL:] is complex-valued ext-real-valued real-valued set
bool [:[:(f),(f):],REAL:] is set
MetrStruct(# (f),(f) #) is strict MetrStruct
TopSpaceMetr (f) is non empty strict TopSpace-like TopStruct
the carrier of (f) is non empty set
Family_open_set (f) is Element of bool (bool the carrier of (f))
bool the carrier of (f) is set
bool (bool the carrier of (f)) is set
TopStruct(# the carrier of (f),(Family_open_set (f)) #) is strict TopStruct
Funcs ((Seg f),REAL) is non empty FUNCTION_DOMAIN of Seg f, REAL
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Element of the carrier of (f)
Seg f is V37() f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= f ) } is set
[:(Seg f),REAL:] is complex-valued ext-real-valued real-valued set
bool [:(Seg f),REAL:] is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Element of the carrier of (f)
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like Element of the carrier of (f)
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like Element of the carrier of (f)
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f + x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
Seg f is V37() f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= f ) } is set
Funcs ((Seg f),REAL) is non empty FUNCTION_DOMAIN of Seg f, REAL
M is Relation-like Function-like complex-valued ext-real-valued real-valued set
M is Relation-like Function-like complex-valued ext-real-valued real-valued set
M + M is Relation-like Function-like complex-valued ext-real-valued real-valued set
the ZeroF of (f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the addF of (f) is Relation-like [: the carrier of (f), the carrier of (f):] -defined the carrier of (f) -valued Function-like quasi_total Element of bool [:[: the carrier of (f), the carrier of (f):], the carrier of (f):]
[: the carrier of (f), the carrier of (f):] is set
[:[: the carrier of (f), the carrier of (f):], the carrier of (f):] is set
bool [:[: the carrier of (f), the carrier of (f):], the carrier of (f):] is set
the Mult of (f) is Relation-like [:REAL, the carrier of (f):] -defined the carrier of (f) -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of (f):], the carrier of (f):]
[:REAL, the carrier of (f):] is V37() set
[:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
bool [:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
RLSStruct(# the carrier of (f), the ZeroF of (f), the addF of (f), the Mult of (f) #) is strict RLSStruct
RealVectSpace (Seg f) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
RealFuncZero (Seg f) is Relation-like Seg f -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of Funcs ((Seg f),REAL)
(Seg f) --> 0 is Relation-like Seg f -defined NAT -valued RAT -valued INT -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of bool [:(Seg f),NAT:]
[:(Seg f),NAT:] is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg f),NAT:] is set
RealFuncAdd (Seg f) is Relation-like [:(Funcs ((Seg f),REAL)),(Funcs ((Seg f),REAL)):] -defined Funcs ((Seg f),REAL) -valued Function-like quasi_total Element of bool [:[:(Funcs ((Seg f),REAL)),(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):]
[:(Funcs ((Seg f),REAL)),(Funcs ((Seg f),REAL)):] is set
[:[:(Funcs ((Seg f),REAL)),(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):] is set
bool [:[:(Funcs ((Seg f),REAL)),(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):] is set
RealFuncExtMult (Seg f) is Relation-like [:REAL,(Funcs ((Seg f),REAL)):] -defined Funcs ((Seg f),REAL) -valued Function-like quasi_total Element of bool [:[:REAL,(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):]
[:REAL,(Funcs ((Seg f),REAL)):] is V37() set
[:[:REAL,(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):] is V37() set
bool [:[:REAL,(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):] is V37() set
RLSStruct(# (Funcs ((Seg f),REAL)),(RealFuncZero (Seg f)),(RealFuncAdd (Seg f)),(RealFuncExtMult (Seg f)) #) is strict RLSStruct
(RealFuncAdd (Seg f)) . (M,M) is set
s is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
f is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
(f,s,f) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
addreal .: (s,f) is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x is complex real ext-real set
x * f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the Mult of (f) is Relation-like [:REAL, the carrier of (f):] -defined the carrier of (f) -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of (f):], the carrier of (f):]
[:REAL, the carrier of (f):] is V37() set
[:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
bool [:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
the Mult of (f) . (x,f) is set
M is Relation-like Function-like complex-valued ext-real-valued real-valued set
x (#) M is Relation-like Function-like complex-valued ext-real-valued real-valued set
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
Seg f is V37() f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= f ) } is set
Funcs ((Seg f),REAL) is non empty FUNCTION_DOMAIN of Seg f, REAL
[:(Seg f),REAL:] is complex-valued ext-real-valued real-valued set
bool [:(Seg f),REAL:] is set
s is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
the ZeroF of (f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the addF of (f) is Relation-like [: the carrier of (f), the carrier of (f):] -defined the carrier of (f) -valued Function-like quasi_total Element of bool [:[: the carrier of (f), the carrier of (f):], the carrier of (f):]
[: the carrier of (f), the carrier of (f):] is set
[:[: the carrier of (f), the carrier of (f):], the carrier of (f):] is set
bool [:[: the carrier of (f), the carrier of (f):], the carrier of (f):] is set
RLSStruct(# the carrier of (f), the ZeroF of (f), the addF of (f), the Mult of (f) #) is strict RLSStruct
RealVectSpace (Seg f) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
RealFuncZero (Seg f) is Relation-like Seg f -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of Funcs ((Seg f),REAL)
(Seg f) --> 0 is Relation-like Seg f -defined NAT -valued RAT -valued INT -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of bool [:(Seg f),NAT:]
[:(Seg f),NAT:] is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg f),NAT:] is set
RealFuncAdd (Seg f) is Relation-like [:(Funcs ((Seg f),REAL)),(Funcs ((Seg f),REAL)):] -defined Funcs ((Seg f),REAL) -valued Function-like quasi_total Element of bool [:[:(Funcs ((Seg f),REAL)),(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):]
[:(Funcs ((Seg f),REAL)),(Funcs ((Seg f),REAL)):] is set
[:[:(Funcs ((Seg f),REAL)),(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):] is set
bool [:[:(Funcs ((Seg f),REAL)),(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):] is set
RealFuncExtMult (Seg f) is Relation-like [:REAL,(Funcs ((Seg f),REAL)):] -defined Funcs ((Seg f),REAL) -valued Function-like quasi_total Element of bool [:[:REAL,(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):]
[:REAL,(Funcs ((Seg f),REAL)):] is V37() set
[:[:REAL,(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):] is V37() set
bool [:[:REAL,(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):] is V37() set
RLSStruct(# (Funcs ((Seg f),REAL)),(RealFuncZero (Seg f)),(RealFuncAdd (Seg f)),(RealFuncExtMult (Seg f)) #) is strict RLSStruct
M is complex real ext-real Element of REAL
f is Relation-like Seg f -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:(Seg f),REAL:]
multreal [;] (M,f) is Relation-like Seg f -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of Funcs ((Seg f),REAL)
id REAL is non empty Relation-like Function-like one-to-one complex-valued ext-real-valued real-valued set
f (#) (id REAL) is Relation-like Function-like complex-valued ext-real-valued real-valued set
multreal [;] (x,(f (#) (id REAL))) is set
(f,s,x) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
x multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] (x,K83(REAL)) is set
s (#) (x multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x is ordinal natural complex real ext-real V37() cardinal set
(x) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (x) is non empty set
f is complex real ext-real set
f is complex real ext-real set
M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (x)
M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f * M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (x)
the Mult of (x) is Relation-like [:REAL, the carrier of (x):] -defined the carrier of (x) -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of (x):], the carrier of (x):]
[:REAL, the carrier of (x):] is V37() set
[:[:REAL, the carrier of (x):], the carrier of (x):] is V37() set
bool [:[:REAL, the carrier of (x):], the carrier of (x):] is V37() set
the Mult of (x) . (f,M) is set
f * M is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] (f,K83(REAL)) is set
M (#) (f multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f * M is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] (f,K83(REAL)) is set
M (#) (f multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
M + M is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (M,M) is set
f + x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (f,x) is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
- f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
x (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) * f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the Mult of (f) is Relation-like [:REAL, the carrier of (f):] -defined the carrier of (f) -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of (f):], the carrier of (f):]
[:REAL, the carrier of (f):] is V37() set
[:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
bool [:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
the Mult of (f) . ((- 1),f) is set
(- 1) * f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(- 1) multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] ((- 1),K83(REAL)) is set
f (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
- f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f - x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
x (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f + (- x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (f,(- x)) is set
M - M is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: (M,M) is set
- M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M + (- M) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (M,(- M)) is set
f - x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: (f,x) is set
- x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (f,(- x)) is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
Seg f is V37() f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= f ) } is set
[:(Seg f),REAL:] is complex-valued ext-real-valued real-valued set
bool [:(Seg f),REAL:] is set
len f is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT
Seg (len f) is V37() len f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= len f ) } is set
dom f is V159() V160() V161() V162() V163() V164() Element of bool NAT
f is ordinal natural complex real ext-real V37() cardinal set
(f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
Seg f is V37() f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= f ) } is set
(Seg f) --> 0 is Relation-like Seg f -defined RAT -valued INT -valued {0} -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of bool [:(Seg f),{0}:]
[:(Seg f),{0}:] is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg f),{0}:] is set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
the topology of (f) is Element of bool (bool the carrier of (f))
bool the carrier of (f) is set
bool (bool the carrier of (f)) is set
TopStruct(# the carrier of (f), the topology of (f) #) is strict TopStruct
(f) is non empty strict Reflexive discerning V86() triangle Discerning MetrStruct
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
(f) is Relation-like [:(f),(f):] -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:[:(f),(f):],REAL:]
[:(f),(f):] is set
[:[:(f),(f):],REAL:] is complex-valued ext-real-valued real-valued set
bool [:[:(f),(f):],REAL:] is set
MetrStruct(# (f),(f) #) is strict MetrStruct
TopSpaceMetr (f) is non empty strict TopSpace-like TopStruct
the carrier of (f) is non empty set
Family_open_set (f) is Element of bool (bool the carrier of (f))
bool the carrier of (f) is set
bool (bool the carrier of (f)) is set
TopStruct(# the carrier of (f),(Family_open_set (f)) #) is strict TopStruct
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
f is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
(f,f) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
f (#) () is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
sqr (f,f) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
(f,f) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,f) (#) (f,f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f,f),(f,f)) is set
(f,f) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
f (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: (f,f) is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f + x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f + x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (f,x) is set
M is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
(f + x) + M is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
(f + x) + M is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: ((f + x),M) is set
x + M is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x + M is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (x,M) is set
f + (x + M) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f + (x + M) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (f,(x + M)) is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
x is ordinal natural complex real ext-real V37() cardinal set
(x) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (x) is non empty set
0. (f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like zero complex-valued ext-real-valued real-valued Element of the carrier of (f)
the ZeroF of (f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
(0. (f)) + f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
(0. (f)) + f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: ((0. (f)),f) is set
M is Relation-like NAT -defined Function-like V37() x -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (x)
0. (x) is Relation-like NAT -defined Function-like V37() x -element FinSequence-like FinSubsequence-like zero complex-valued ext-real-valued real-valued Element of the carrier of (x)
the ZeroF of (x) is Relation-like NAT -defined Function-like V37() x -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (x)
M + (0. (x)) is Relation-like NAT -defined Function-like V37() x -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (x)
M + (0. (x)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (M,(0. (x))) is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
0. (f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like zero complex-valued ext-real-valued real-valued Element of the carrier of (f)
the carrier of (f) is non empty set
the ZeroF of (f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f is complex real ext-real set
f * (0. (f)) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the Mult of (f) is Relation-like [:REAL, the carrier of (f):] -defined the carrier of (f) -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of (f):], the carrier of (f):]
[:REAL, the carrier of (f):] is V37() set
[:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
bool [:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
the Mult of (f) . (f,(0. (f))) is set
f * (0. (f)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] (f,K83(REAL)) is set
(0. (f)) (#) (f multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
x is ordinal natural complex real ext-real V37() cardinal set
(x) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (x) is non empty set
f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
1 * f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the Mult of (f) is Relation-like [:REAL, the carrier of (f):] -defined the carrier of (f) -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of (f):], the carrier of (f):]
[:REAL, the carrier of (f):] is V37() set
[:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
bool [:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
the Mult of (f) . (1,f) is set
1 * f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
1 multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] (1,K83(REAL)) is set
f (#) (1 multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M is Relation-like NAT -defined Function-like V37() x -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (x)
0 * M is Relation-like NAT -defined Function-like V37() x -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (x)
the Mult of (x) is Relation-like [:REAL, the carrier of (x):] -defined the carrier of (x) -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of (x):], the carrier of (x):]
[:REAL, the carrier of (x):] is V37() set
[:[:REAL, the carrier of (x):], the carrier of (x):] is V37() set
bool [:[:REAL, the carrier of (x):], the carrier of (x):] is V37() set
the Mult of (x) . (0,M) is set
0 * M is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
0 multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] (0,K83(REAL)) is set
M (#) (0 multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
0. (x) is Relation-like NAT -defined Function-like V37() x -element FinSequence-like FinSubsequence-like zero complex-valued ext-real-valued real-valued Element of the carrier of (x)
the ZeroF of (x) is Relation-like NAT -defined Function-like V37() x -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (x)
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x is complex real ext-real set
M is complex real ext-real set
x * M is complex real ext-real Element of REAL
(x * M) * f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the Mult of (f) is Relation-like [:REAL, the carrier of (f):] -defined the carrier of (f) -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of (f):], the carrier of (f):]
[:REAL, the carrier of (f):] is V37() set
[:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
bool [:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
the Mult of (f) . ((x * M),f) is set
(x * M) * f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(x * M) multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] ((x * M),K83(REAL)) is set
f (#) ((x * M) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M * f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the Mult of (f) . (M,f) is set
M * f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
M multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] (M,K83(REAL)) is set
f (#) (M multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x * (M * f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the Mult of (f) . (x,(M * f)) is set
x * (M * f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
x multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] (x,K83(REAL)) is set
(M * f) (#) (x multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x is complex real ext-real set
x * f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the Mult of (f) is Relation-like [:REAL, the carrier of (f):] -defined the carrier of (f) -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of (f):], the carrier of (f):]
[:REAL, the carrier of (f):] is V37() set
[:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
bool [:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
the Mult of (f) . (x,f) is set
x * f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
x multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] (x,K83(REAL)) is set
f (#) (x multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
0. (f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like zero complex-valued ext-real-valued real-valued Element of the carrier of (f)
the ZeroF of (f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f + x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f + x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (f,x) is set
M is complex real ext-real set
M * (f + x) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the Mult of (f) is Relation-like [:REAL, the carrier of (f):] -defined the carrier of (f) -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of (f):], the carrier of (f):]
[:REAL, the carrier of (f):] is V37() set
[:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
bool [:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
the Mult of (f) . (M,(f + x)) is set
M * (f + x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
M multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] (M,K83(REAL)) is set
(f + x) (#) (M multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M * f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the Mult of (f) . (M,f) is set
M * f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f (#) (M multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M * x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the Mult of (f) . (M,x) is set
M * x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
x (#) (M multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(M * f) + (M * x) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
(M * f) + (M * x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: ((M * f),(M * x)) is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x is complex real ext-real set
M is complex real ext-real set
x + M is complex real ext-real Element of REAL
(x + M) * f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the Mult of (f) is Relation-like [:REAL, the carrier of (f):] -defined the carrier of (f) -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of (f):], the carrier of (f):]
[:REAL, the carrier of (f):] is V37() set
[:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
bool [:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
the Mult of (f) . ((x + M),f) is set
(x + M) * f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(x + M) multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] ((x + M),K83(REAL)) is set
f (#) ((x + M) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x * f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the Mult of (f) . (x,f) is set
x * f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
x multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] (x,K83(REAL)) is set
f (#) (x multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M * f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the Mult of (f) . (M,f) is set
M * f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
M multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] (M,K83(REAL)) is set
f (#) (M multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(x * f) + (M * f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
(x * f) + (M * f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: ((x * f),(M * f)) is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
M is complex real ext-real set
M * f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the Mult of (f) is Relation-like [:REAL, the carrier of (f):] -defined the carrier of (f) -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of (f):], the carrier of (f):]
[:REAL, the carrier of (f):] is V37() set
[:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
bool [:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
the Mult of (f) . (M,f) is set
M * f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
M multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] (M,K83(REAL)) is set
f (#) (M multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
M * x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the Mult of (f) . (M,x) is set
M * x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
x (#) (M multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f + (- f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (f,(- f)) is set
0. (f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like zero complex-valued ext-real-valued real-valued Element of the carrier of (f)
the ZeroF of (f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f + x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f + x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (f,x) is set
0. (f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like zero complex-valued ext-real-valued real-valued Element of the carrier of (f)
the ZeroF of (f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
x (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f + x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f + x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (f,x) is set
- (f + x) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- (f + x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f + x) (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) (f + x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f + x) (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
- f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- f) - x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
x (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- f) + (- x) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
(- f) + (- x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: ((- f),(- x)) is set
(- f) - x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: ((- f),x) is set
- x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- f) + (- x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: ((- f),(- x)) is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) * f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the Mult of (f) is Relation-like [:REAL, the carrier of (f):] -defined the carrier of (f) -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of (f):], the carrier of (f):]
[:REAL, the carrier of (f):] is V37() set
[:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
bool [:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
the Mult of (f) . ((- 1),f) is set
(- 1) * f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(- 1) multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] ((- 1),K83(REAL)) is set
f (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
M is ordinal natural complex real ext-real V37() cardinal set
(M) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (M) is non empty set
f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x is complex real ext-real set
x * f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the Mult of (f) is Relation-like [:REAL, the carrier of (f):] -defined the carrier of (f) -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of (f):], the carrier of (f):]
[:REAL, the carrier of (f):] is V37() set
[:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
bool [:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
the Mult of (f) . (x,f) is set
x * f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
x multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] (x,K83(REAL)) is set
f (#) (x multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
- (x * f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- (x * f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(x * f) (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) (x * f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(x * f) (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
- x is complex real ext-real Element of REAL
(- x) * f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the Mult of (f) . ((- x),f) is set
(- x) * f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(- x) multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] ((- x),K83(REAL)) is set
f (#) ((- x) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M is Relation-like NAT -defined Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (M)
s is complex real ext-real set
s * M is Relation-like NAT -defined Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (M)
the Mult of (M) is Relation-like [:REAL, the carrier of (M):] -defined the carrier of (M) -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of (M):], the carrier of (M):]
[:REAL, the carrier of (M):] is V37() set
[:[:REAL, the carrier of (M):], the carrier of (M):] is V37() set
bool [:[:REAL, the carrier of (M):], the carrier of (M):] is V37() set
the Mult of (M) . (s,M) is set
s * M is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
s multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] (s,K83(REAL)) is set
M (#) (s multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
- (s * M) is Relation-like NAT -defined Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (M)
- (s * M) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(s * M) (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) (s * M) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(s * M) (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
- M is Relation-like NAT -defined Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (M)
- M is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
M (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
s * (- M) is Relation-like NAT -defined Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (M)
the Mult of (M) . (s,(- M)) is set
s * (- M) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(- M) (#) (s multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f - x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
x (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- x) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f + (- x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (f,(- x)) is set
f - x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: (f,x) is set
- x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (f,(- x)) is set
f + (- x) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f - f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f + (- f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (f,(- f)) is set
f - f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: (f,f) is set
- f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (f,(- f)) is set
0. (f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like zero complex-valued ext-real-valued real-valued Element of the carrier of (f)
the ZeroF of (f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f - x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
x (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- x) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f + (- x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (f,(- x)) is set
f - x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: (f,x) is set
- x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (f,(- x)) is set
0. (f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like zero complex-valued ext-real-valued real-valued Element of the carrier of (f)
the ZeroF of (f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
M is ordinal natural complex real ext-real V37() cardinal set
(M) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (M) is non empty set
f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f - x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
x (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- x) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f + (- x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (f,(- x)) is set
f - x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: (f,x) is set
- x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (f,(- x)) is set
- (f - x) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- (f - x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f - x) (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) (f - x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f - x) (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x - f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x + (- f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x + (- f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (x,(- f)) is set
x - f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: (x,f) is set
- f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x + (- f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (x,(- f)) is set
M is Relation-like NAT -defined Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (M)
s is Relation-like NAT -defined Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (M)
M - s is Relation-like NAT -defined Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (M)
- s is Relation-like NAT -defined Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (M)
- s is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
s (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) s is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
s (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M + (- s) is Relation-like NAT -defined Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (M)
M + (- s) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (M,(- s)) is set
M - s is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: (M,s) is set
- s is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M + (- s) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (M,(- s)) is set
- (M - s) is Relation-like NAT -defined Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (M)
- (M - s) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(M - s) (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) (M - s) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(M - s) (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
- M is Relation-like NAT -defined Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (M)
- M is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
M (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- M) + s is Relation-like NAT -defined Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (M)
(- M) + s is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: ((- M),s) is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
M is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x - M is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- M is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- M is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
M (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x + (- M) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x + (- M) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (x,(- M)) is set
x - M is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: (x,M) is set
- M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x + (- M) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (x,(- M)) is set
f + (x - M) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f + (x - M) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (f,(x - M)) is set
f + x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f + x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (f,x) is set
(f + x) - M is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
(f + x) + (- M) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
(f + x) + (- M) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: ((f + x),(- M)) is set
(f + x) - M is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: ((f + x),M) is set
(f + x) + (- M) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: ((f + x),(- M)) is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
M is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x + M is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x + M is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (x,M) is set
f - (x + M) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- (x + M) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- (x + M) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(x + M) (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) (x + M) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(x + M) (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- (x + M)) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f + (- (x + M)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (f,(- (x + M))) is set
f - (x + M) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: (f,(x + M)) is set
- (x + M) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- (x + M)) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (f,(- (x + M))) is set
f - x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
x (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- x) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f + (- x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (f,(- x)) is set
f - x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: (f,x) is set
- x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (f,(- x)) is set
(f - x) - M is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- M is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- M is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
M (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f - x) + (- M) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
(f - x) + (- M) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: ((f - x),(- M)) is set
(f - x) - M is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: ((f - x),M) is set
- M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f - x) + (- M) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: ((f - x),(- M)) is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
M is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x - M is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- M is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- M is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
M (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x + (- M) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x + (- M) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (x,(- M)) is set
x - M is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: (x,M) is set
- M is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x + (- M) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (x,(- M)) is set
f - (x - M) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- (x - M) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- (x - M) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(x - M) (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) (x - M) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(x - M) (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- (x - M)) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f + (- (x - M)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (f,(- (x - M))) is set
f - (x - M) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: (f,(x - M)) is set
- (x - M) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- (x - M)) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (f,(- (x - M))) is set
f - x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
x (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- x) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f + (- x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (f,(- x)) is set
f - x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: (f,x) is set
- x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (f,(- x)) is set
(f - x) + M is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
(f - x) + M is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: ((f - x),M) is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
M is ordinal natural complex real ext-real V37() cardinal set
(M) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (M) is non empty set
f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f + x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f + x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (f,x) is set
(f + x) - x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
x (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f + x) + (- x) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
(f + x) + (- x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: ((f + x),(- x)) is set
(f + x) - x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: ((f + x),x) is set
- x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f + x) + (- x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: ((f + x),(- x)) is set
M is Relation-like NAT -defined Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (M)
s is Relation-like NAT -defined Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (M)
M - s is Relation-like NAT -defined Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (M)
- s is Relation-like NAT -defined Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (M)
- s is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
s (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) s is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
s (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M + (- s) is Relation-like NAT -defined Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (M)
M + (- s) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (M,(- s)) is set
M - s is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: (M,s) is set
- s is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M + (- s) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (M,(- s)) is set
(M - s) + s is Relation-like NAT -defined Function-like V37() M -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (M)
(M - s) + s is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: ((M - s),s) is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f - x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
x (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- x) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f + (- x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (f,(- x)) is set
f - x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: (f,x) is set
- x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (f,(- x)) is set
M is complex real ext-real set
M * (f - x) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the Mult of (f) is Relation-like [:REAL, the carrier of (f):] -defined the carrier of (f) -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of (f):], the carrier of (f):]
[:REAL, the carrier of (f):] is V37() set
[:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
bool [:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
the Mult of (f) . (M,(f - x)) is set
M * (f - x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
M multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] (M,K83(REAL)) is set
(f - x) (#) (M multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M * f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the Mult of (f) . (M,f) is set
M * f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f (#) (M multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M * x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the Mult of (f) . (M,x) is set
M * x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
x (#) (M multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(M * f) - (M * x) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- (M * x) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- (M * x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(M * x) (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) (M * x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(M * x) (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(M * f) + (- (M * x)) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
(M * f) + (- (M * x)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: ((M * f),(- (M * x))) is set
(M * f) - (M * x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: ((M * f),(M * x)) is set
- (M * x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(M * f) + (- (M * x)) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: ((M * f),(- (M * x))) is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x is complex real ext-real set
M is complex real ext-real set
x - M is complex real ext-real Element of REAL
(x - M) * f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the Mult of (f) is Relation-like [:REAL, the carrier of (f):] -defined the carrier of (f) -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of (f):], the carrier of (f):]
[:REAL, the carrier of (f):] is V37() set
[:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
bool [:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
the Mult of (f) . ((x - M),f) is set
(x - M) * f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(x - M) multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] ((x - M),K83(REAL)) is set
f (#) ((x - M) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x * f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the Mult of (f) . (x,f) is set
x * f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
x multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] (x,K83(REAL)) is set
f (#) (x multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
M * f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the Mult of (f) . (M,f) is set
M * f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
M multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] (M,K83(REAL)) is set
f (#) (M multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(x * f) - (M * f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- (M * f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- (M * f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(M * f) (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) (M * f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(M * f) (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(x * f) + (- (M * f)) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
(x * f) + (- (M * f)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: ((x * f),(- (M * f))) is set
(x * f) - (M * f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: ((x * f),(M * f)) is set
- (M * f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(x * f) + (- (M * f)) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: ((x * f),(- (M * f))) is set
(2) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (2) is non empty set
f is Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
the topology of (2) is Element of bool (bool the carrier of (2))
bool the carrier of (2) is set
bool (bool the carrier of (2)) is set
TopStruct(# the carrier of (2), the topology of (2) #) is strict TopStruct
(2) is non empty strict Reflexive discerning V86() triangle Discerning MetrStruct
(2) is non empty functional FinSequence-membered FinSequenceSet of REAL
2 -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = 2 } is set
(2) is Relation-like [:(2),(2):] -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:[:(2),(2):],REAL:]
[:(2),(2):] is set
[:[:(2),(2):],REAL:] is complex-valued ext-real-valued real-valued set
bool [:[:(2),(2):],REAL:] is set
MetrStruct(# (2),(2) #) is strict MetrStruct
TopSpaceMetr (2) is non empty strict TopSpace-like TopStruct
the carrier of (2) is non empty set
Family_open_set (2) is Element of bool (bool the carrier of (2))
bool the carrier of (2) is set
bool (bool the carrier of (2)) is set
TopStruct(# the carrier of (2),(Family_open_set (2)) #) is strict TopStruct
f is Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
f . 1 is complex real ext-real Element of REAL
f . 2 is complex real ext-real Element of REAL
f is complex real ext-real set
f is complex real ext-real set
<*f,f*> is non empty Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
<*f*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,f] is set
{[1,f]} is non empty trivial Function-like 1 -element set
<*f*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,f] is set
{[1,f]} is non empty trivial Function-like 1 -element set
K230(<*f*>,<*f*>) is non empty Relation-like NAT -defined Function-like V37() 1 + 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
1 + 1 is non empty ordinal natural complex real ext-real positive non negative V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT
the topology of (2) is Element of bool (bool the carrier of (2))
bool the carrier of (2) is set
bool (bool the carrier of (2)) is set
TopStruct(# the carrier of (2), the topology of (2) #) is strict TopStruct
(2) is non empty strict Reflexive discerning V86() triangle Discerning MetrStruct
(2) is non empty functional FinSequence-membered FinSequenceSet of REAL
2 -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = 2 } is set
(2) is Relation-like [:(2),(2):] -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:[:(2),(2):],REAL:]
[:(2),(2):] is set
[:[:(2),(2):],REAL:] is complex-valued ext-real-valued real-valued set
bool [:[:(2),(2):],REAL:] is set
MetrStruct(# (2),(2) #) is strict MetrStruct
TopSpaceMetr (2) is non empty strict TopSpace-like TopStruct
the carrier of (2) is non empty set
Family_open_set (2) is Element of bool (bool the carrier of (2))
bool the carrier of (2) is set
bool (bool the carrier of (2)) is set
TopStruct(# the carrier of (2),(Family_open_set (2)) #) is strict TopStruct
f is complex real ext-real set
f is complex real ext-real set
(f,f) is non empty Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
<*f*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,f] is set
{[1,f]} is non empty trivial Function-like 1 -element set
<*f*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,f] is set
{[1,f]} is non empty trivial Function-like 1 -element set
K230(<*f*>,<*f*>) is non empty Relation-like NAT -defined Function-like V37() 1 + 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((f,f)) is complex real ext-real Element of REAL
(f,f) . 1 is complex real ext-real Element of REAL
x is complex real ext-real set
M is complex real ext-real set
(x,M) is non empty Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
<*x*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,x] is set
{[1,x]} is non empty trivial Function-like 1 -element set
<*M*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,M] is set
{[1,M]} is non empty trivial Function-like 1 -element set
K230(<*x*>,<*M*>) is non empty Relation-like NAT -defined Function-like V37() 1 + 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((x,M)) is complex real ext-real Element of REAL
(x,M) . 2 is complex real ext-real Element of REAL
f is Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
(f) is complex real ext-real Element of REAL
f . 1 is complex real ext-real Element of REAL
(f) is complex real ext-real Element of REAL
f . 2 is complex real ext-real Element of REAL
((f),(f)) is non empty Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
<*(f)*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,(f)] is set
{[1,(f)]} is non empty trivial Function-like 1 -element set
<*(f)*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,(f)] is set
{[1,(f)]} is non empty trivial Function-like 1 -element set
K230(<*(f)*>,<*(f)*>) is non empty Relation-like NAT -defined Function-like V37() 1 + 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
the topology of (2) is Element of bool (bool the carrier of (2))
bool the carrier of (2) is set
bool (bool the carrier of (2)) is set
TopStruct(# the carrier of (2), the topology of (2) #) is strict TopStruct
(2) is non empty strict Reflexive discerning V86() triangle Discerning MetrStruct
(2) is non empty functional FinSequence-membered FinSequenceSet of REAL
2 -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = 2 } is set
(2) is Relation-like [:(2),(2):] -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:[:(2),(2):],REAL:]
[:(2),(2):] is set
[:[:(2),(2):],REAL:] is complex-valued ext-real-valued real-valued set
bool [:[:(2),(2):],REAL:] is set
MetrStruct(# (2),(2) #) is strict MetrStruct
TopSpaceMetr (2) is non empty strict TopSpace-like TopStruct
the carrier of (2) is non empty set
Family_open_set (2) is Element of bool (bool the carrier of (2))
bool the carrier of (2) is set
bool (bool the carrier of (2)) is set
TopStruct(# the carrier of (2),(Family_open_set (2)) #) is strict TopStruct
f is complex real ext-real Element of REAL
x is complex real ext-real Element of REAL
<*f,x*> is non empty Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
<*f*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,f] is set
{[1,f]} is non empty trivial Function-like 1 -element set
<*x*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,x] is set
{[1,x]} is non empty trivial Function-like 1 -element set
K230(<*f*>,<*x*>) is non empty Relation-like NAT -defined Function-like V37() 1 + 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
0. (2) is Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like zero complex-valued ext-real-valued real-valued Element of the carrier of (2)
the ZeroF of (2) is Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
(0,0) is non empty Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
<*0*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,0] is set
{[1,0]} is non empty trivial Function-like 1 -element set
K230(<*0*>,<*0*>) is non empty Relation-like NAT -defined Function-like V37() 1 + 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
the addF of (2) is Relation-like [: the carrier of (2), the carrier of (2):] -defined the carrier of (2) -valued Function-like quasi_total Element of bool [:[: the carrier of (2), the carrier of (2):], the carrier of (2):]
[: the carrier of (2), the carrier of (2):] is set
[:[: the carrier of (2), the carrier of (2):], the carrier of (2):] is set
bool [:[: the carrier of (2), the carrier of (2):], the carrier of (2):] is set
the Mult of (2) is Relation-like [:REAL, the carrier of (2):] -defined the carrier of (2) -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of (2):], the carrier of (2):]
[:REAL, the carrier of (2):] is V37() set
[:[:REAL, the carrier of (2):], the carrier of (2):] is V37() set
bool [:[:REAL, the carrier of (2):], the carrier of (2):] is V37() set
RLSStruct(# the carrier of (2), the ZeroF of (2), the addF of (2), the Mult of (2) #) is strict RLSStruct
RealVectSpace (Seg 2) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
Funcs ((Seg 2),REAL) is non empty FUNCTION_DOMAIN of Seg 2, REAL
RealFuncZero (Seg 2) is Relation-like Seg 2 -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of Funcs ((Seg 2),REAL)
(Seg 2) --> 0 is Relation-like Seg 2 -defined NAT -valued RAT -valued INT -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of bool [:(Seg 2),NAT:]
[:(Seg 2),NAT:] is RAT -valued INT -valued V37() complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg 2),NAT:] is V37() set
RealFuncAdd (Seg 2) is Relation-like [:(Funcs ((Seg 2),REAL)),(Funcs ((Seg 2),REAL)):] -defined Funcs ((Seg 2),REAL) -valued Function-like quasi_total Element of bool [:[:(Funcs ((Seg 2),REAL)),(Funcs ((Seg 2),REAL)):],(Funcs ((Seg 2),REAL)):]
[:(Funcs ((Seg 2),REAL)),(Funcs ((Seg 2),REAL)):] is set
[:[:(Funcs ((Seg 2),REAL)),(Funcs ((Seg 2),REAL)):],(Funcs ((Seg 2),REAL)):] is set
bool [:[:(Funcs ((Seg 2),REAL)),(Funcs ((Seg 2),REAL)):],(Funcs ((Seg 2),REAL)):] is set
RealFuncExtMult (Seg 2) is Relation-like [:REAL,(Funcs ((Seg 2),REAL)):] -defined Funcs ((Seg 2),REAL) -valued Function-like quasi_total Element of bool [:[:REAL,(Funcs ((Seg 2),REAL)):],(Funcs ((Seg 2),REAL)):]
[:REAL,(Funcs ((Seg 2),REAL)):] is V37() set
[:[:REAL,(Funcs ((Seg 2),REAL)):],(Funcs ((Seg 2),REAL)):] is V37() set
bool [:[:REAL,(Funcs ((Seg 2),REAL)):],(Funcs ((Seg 2),REAL)):] is V37() set
RLSStruct(# (Funcs ((Seg 2),REAL)),(RealFuncZero (Seg 2)),(RealFuncAdd (Seg 2)),(RealFuncExtMult (Seg 2)) #) is strict RLSStruct
(2) is Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
2 |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of 2 -tuples_on REAL
2 -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = 2 } is set
(Seg 2) --> 0 is Relation-like Seg 2 -defined RAT -valued INT -valued {0} -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of bool [:(Seg 2),{0}:]
[:(Seg 2),{0}:] is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg 2),{0}:] is set
f is Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
(f) is complex real ext-real Element of REAL
f . 1 is complex real ext-real Element of REAL
(f) is complex real ext-real Element of REAL
f . 2 is complex real ext-real Element of REAL
f is Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
f + f is Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
f + f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (f,f) is set
(f) is complex real ext-real Element of REAL
f . 1 is complex real ext-real Element of REAL
(f) + (f) is complex real ext-real Element of REAL
(f) is complex real ext-real Element of REAL
f . 2 is complex real ext-real Element of REAL
(f) + (f) is complex real ext-real Element of REAL
(((f) + (f)),((f) + (f))) is non empty Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
<*((f) + (f))*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,((f) + (f))] is set
{[1,((f) + (f))]} is non empty trivial Function-like 1 -element set
<*((f) + (f))*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,((f) + (f))] is set
{[1,((f) + (f))]} is non empty trivial Function-like 1 -element set
K230(<*((f) + (f))*>,<*((f) + (f))*>) is non empty Relation-like NAT -defined Function-like V37() 1 + 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
the topology of (2) is Element of bool (bool the carrier of (2))
bool the carrier of (2) is set
bool (bool the carrier of (2)) is set
TopStruct(# the carrier of (2), the topology of (2) #) is strict TopStruct
(2) is non empty strict Reflexive discerning V86() triangle Discerning MetrStruct
(2) is non empty functional FinSequence-membered FinSequenceSet of REAL
2 -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = 2 } is set
(2) is Relation-like [:(2),(2):] -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:[:(2),(2):],REAL:]
[:(2),(2):] is set
[:[:(2),(2):],REAL:] is complex-valued ext-real-valued real-valued set
bool [:[:(2),(2):],REAL:] is set
MetrStruct(# (2),(2) #) is strict MetrStruct
TopSpaceMetr (2) is non empty strict TopSpace-like TopStruct
the carrier of (2) is non empty set
Family_open_set (2) is Element of bool (bool the carrier of (2))
bool the carrier of (2) is set
bool (bool the carrier of (2)) is set
TopStruct(# the carrier of (2),(Family_open_set (2)) #) is strict TopStruct
x is Relation-like NAT -defined REAL -valued Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (2)
M is Relation-like NAT -defined REAL -valued Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (2)
(2,x,M) is Relation-like NAT -defined REAL -valued Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (2)
addreal .: (x,M) is set
len (2,x,M) is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT
dom (2,x,M) is 2 -element V159() V160() V161() V162() V163() V164() Element of bool NAT
((f + f)) is complex real ext-real Element of REAL
(f + f) . 2 is complex real ext-real Element of REAL
((f + f)) is complex real ext-real Element of REAL
(f + f) . 1 is complex real ext-real Element of REAL
f is complex real ext-real set
f is complex real ext-real set
(f,f) is non empty Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
<*f*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,f] is set
{[1,f]} is non empty trivial Function-like 1 -element set
<*f*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,f] is set
{[1,f]} is non empty trivial Function-like 1 -element set
K230(<*f*>,<*f*>) is non empty Relation-like NAT -defined Function-like V37() 1 + 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x is complex real ext-real set
f + x is complex real ext-real Element of REAL
M is complex real ext-real set
(x,M) is non empty Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
<*x*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,x] is set
{[1,x]} is non empty trivial Function-like 1 -element set
<*M*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,M] is set
{[1,M]} is non empty trivial Function-like 1 -element set
K230(<*x*>,<*M*>) is non empty Relation-like NAT -defined Function-like V37() 1 + 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,f) + (x,M) is Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
(f,f) + (x,M) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: ((f,f),(x,M)) is set
f + M is complex real ext-real Element of REAL
((f + x),(f + M)) is non empty Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
<*(f + x)*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,(f + x)] is set
{[1,(f + x)]} is non empty trivial Function-like 1 -element set
<*(f + M)*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,(f + M)] is set
{[1,(f + M)]} is non empty trivial Function-like 1 -element set
K230(<*(f + x)*>,<*(f + M)*>) is non empty Relation-like NAT -defined Function-like V37() 1 + 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((x,M)) is complex real ext-real Element of REAL
(x,M) . 1 is complex real ext-real Element of REAL
((x,M)) is complex real ext-real Element of REAL
(x,M) . 2 is complex real ext-real Element of REAL
((f,f)) is complex real ext-real Element of REAL
(f,f) . 1 is complex real ext-real Element of REAL
((f,f)) is complex real ext-real Element of REAL
(f,f) . 2 is complex real ext-real Element of REAL
f is complex real ext-real set
f is Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
f * f is Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
the Mult of (2) is Relation-like [:REAL, the carrier of (2):] -defined the carrier of (2) -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of (2):], the carrier of (2):]
[:REAL, the carrier of (2):] is V37() set
[:[:REAL, the carrier of (2):], the carrier of (2):] is V37() set
bool [:[:REAL, the carrier of (2):], the carrier of (2):] is V37() set
the Mult of (2) . (f,f) is set
f * f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] (f,K83(REAL)) is set
f (#) (f multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f) is complex real ext-real Element of REAL
f . 1 is complex real ext-real Element of REAL
f * (f) is complex real ext-real Element of REAL
(f) is complex real ext-real Element of REAL
f . 2 is complex real ext-real Element of REAL
f * (f) is complex real ext-real Element of REAL
((f * (f)),(f * (f))) is non empty Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
<*(f * (f))*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,(f * (f))] is set
{[1,(f * (f))]} is non empty trivial Function-like 1 -element set
<*(f * (f))*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,(f * (f))] is set
{[1,(f * (f))]} is non empty trivial Function-like 1 -element set
K230(<*(f * (f))*>,<*(f * (f))*>) is non empty Relation-like NAT -defined Function-like V37() 1 + 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((f * f)) is complex real ext-real Element of REAL
(f * f) . 1 is complex real ext-real Element of REAL
((f * f)) is complex real ext-real Element of REAL
(f * f) . 2 is complex real ext-real Element of REAL
f is complex real ext-real set
f is complex real ext-real set
f * f is complex real ext-real Element of REAL
x is complex real ext-real set
(f,x) is non empty Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
<*f*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,f] is set
{[1,f]} is non empty trivial Function-like 1 -element set
<*x*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,x] is set
{[1,x]} is non empty trivial Function-like 1 -element set
K230(<*f*>,<*x*>) is non empty Relation-like NAT -defined Function-like V37() 1 + 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f * (f,x) is Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
the Mult of (2) is Relation-like [:REAL, the carrier of (2):] -defined the carrier of (2) -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of (2):], the carrier of (2):]
[:REAL, the carrier of (2):] is V37() set
[:[:REAL, the carrier of (2):], the carrier of (2):] is V37() set
bool [:[:REAL, the carrier of (2):], the carrier of (2):] is V37() set
the Mult of (2) . (f,(f,x)) is set
f * (f,x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] (f,K83(REAL)) is set
(f,x) (#) (f multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f * x is complex real ext-real Element of REAL
((f * f),(f * x)) is non empty Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
<*(f * f)*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,(f * f)] is set
{[1,(f * f)]} is non empty trivial Function-like 1 -element set
<*(f * x)*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,(f * x)] is set
{[1,(f * x)]} is non empty trivial Function-like 1 -element set
K230(<*(f * f)*>,<*(f * x)*>) is non empty Relation-like NAT -defined Function-like V37() 1 + 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((f,x)) is complex real ext-real Element of REAL
(f,x) . 1 is complex real ext-real Element of REAL
((f,x)) is complex real ext-real Element of REAL
(f,x) . 2 is complex real ext-real Element of REAL
f is Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
- f is Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
- f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f) is complex real ext-real Element of REAL
f . 1 is complex real ext-real Element of REAL
- (f) is complex real ext-real Element of REAL
(f) is complex real ext-real Element of REAL
f . 2 is complex real ext-real Element of REAL
- (f) is complex real ext-real Element of REAL
((- (f)),(- (f))) is non empty Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
<*(- (f))*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,(- (f))] is set
{[1,(- (f))]} is non empty trivial Function-like 1 -element set
<*(- (f))*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,(- (f))] is set
{[1,(- (f))]} is non empty trivial Function-like 1 -element set
K230(<*(- (f))*>,<*(- (f))*>) is non empty Relation-like NAT -defined Function-like V37() 1 + 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) * f is Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
the Mult of (2) is Relation-like [:REAL, the carrier of (2):] -defined the carrier of (2) -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of (2):], the carrier of (2):]
[:REAL, the carrier of (2):] is V37() set
[:[:REAL, the carrier of (2):], the carrier of (2):] is V37() set
bool [:[:REAL, the carrier of (2):], the carrier of (2):] is V37() set
the Mult of (2) . ((- 1),f) is set
(- 1) * f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(- 1) multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] ((- 1),K83(REAL)) is set
f (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) * (f) is complex real ext-real Element of REAL
(- 1) * (f) is complex real ext-real Element of REAL
(((- 1) * (f)),((- 1) * (f))) is non empty Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
<*((- 1) * (f))*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,((- 1) * (f))] is set
{[1,((- 1) * (f))]} is non empty trivial Function-like 1 -element set
<*((- 1) * (f))*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,((- 1) * (f))] is set
{[1,((- 1) * (f))]} is non empty trivial Function-like 1 -element set
K230(<*((- 1) * (f))*>,<*((- 1) * (f))*>) is non empty Relation-like NAT -defined Function-like V37() 1 + 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f is complex real ext-real set
- f is complex real ext-real Element of REAL
f is complex real ext-real set
(f,f) is non empty Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
<*f*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,f] is set
{[1,f]} is non empty trivial Function-like 1 -element set
<*f*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,f] is set
{[1,f]} is non empty trivial Function-like 1 -element set
K230(<*f*>,<*f*>) is non empty Relation-like NAT -defined Function-like V37() 1 + 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
- (f,f) is Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
- (f,f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f,f) (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) (f,f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,f) (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
- f is complex real ext-real Element of REAL
((- f),(- f)) is non empty Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
<*(- f)*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,(- f)] is set
{[1,(- f)]} is non empty trivial Function-like 1 -element set
<*(- f)*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,(- f)] is set
{[1,(- f)]} is non empty trivial Function-like 1 -element set
K230(<*(- f)*>,<*(- f)*>) is non empty Relation-like NAT -defined Function-like V37() 1 + 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((f,f)) is complex real ext-real Element of REAL
(f,f) . 1 is complex real ext-real Element of REAL
((f,f)) is complex real ext-real Element of REAL
(f,f) . 2 is complex real ext-real Element of REAL
f is Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
(f) is complex real ext-real Element of REAL
f . 1 is complex real ext-real Element of REAL
(f) is complex real ext-real Element of REAL
f . 2 is complex real ext-real Element of REAL
f is Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
f - f is Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
- f is Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
- f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- f) is Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
f + (- f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (f,(- f)) is set
f - f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: (f,f) is set
- f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (f,(- f)) is set
(f) is complex real ext-real Element of REAL
f . 1 is complex real ext-real Element of REAL
(f) - (f) is complex real ext-real Element of REAL
(f) is complex real ext-real Element of REAL
f . 2 is complex real ext-real Element of REAL
(f) - (f) is complex real ext-real Element of REAL
(((f) - (f)),((f) - (f))) is non empty Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
<*((f) - (f))*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,((f) - (f))] is set
{[1,((f) - (f))]} is non empty trivial Function-like 1 -element set
<*((f) - (f))*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,((f) - (f))] is set
{[1,((f) - (f))]} is non empty trivial Function-like 1 -element set
K230(<*((f) - (f))*>,<*((f) - (f))*>) is non empty Relation-like NAT -defined Function-like V37() 1 + 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
- (f) is complex real ext-real Element of REAL
- (f) is complex real ext-real Element of REAL
((- (f)),(- (f))) is non empty Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
<*(- (f))*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,(- (f))] is set
{[1,(- (f))]} is non empty trivial Function-like 1 -element set
<*(- (f))*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,(- (f))] is set
{[1,(- (f))]} is non empty trivial Function-like 1 -element set
K230(<*(- (f))*>,<*(- (f))*>) is non empty Relation-like NAT -defined Function-like V37() 1 + 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((- f)) is complex real ext-real Element of REAL
(- f) . 1 is complex real ext-real Element of REAL
((- f)) is complex real ext-real Element of REAL
(- f) . 2 is complex real ext-real Element of REAL
(f) + (- (f)) is complex real ext-real Element of REAL
(f) + (- (f)) is complex real ext-real Element of REAL
(((f) + (- (f))),((f) + (- (f)))) is non empty Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
<*((f) + (- (f)))*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,((f) + (- (f)))] is set
{[1,((f) + (- (f)))]} is non empty trivial Function-like 1 -element set
<*((f) + (- (f)))*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,((f) + (- (f)))] is set
{[1,((f) + (- (f)))]} is non empty trivial Function-like 1 -element set
K230(<*((f) + (- (f)))*>,<*((f) + (- (f)))*>) is non empty Relation-like NAT -defined Function-like V37() 1 + 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f is complex real ext-real set
f is complex real ext-real set
(f,f) is non empty Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
<*f*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,f] is set
{[1,f]} is non empty trivial Function-like 1 -element set
<*f*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,f] is set
{[1,f]} is non empty trivial Function-like 1 -element set
K230(<*f*>,<*f*>) is non empty Relation-like NAT -defined Function-like V37() 1 + 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x is complex real ext-real set
f - x is complex real ext-real Element of REAL
M is complex real ext-real set
(x,M) is non empty Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
<*x*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,x] is set
{[1,x]} is non empty trivial Function-like 1 -element set
<*M*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,M] is set
{[1,M]} is non empty trivial Function-like 1 -element set
K230(<*x*>,<*M*>) is non empty Relation-like NAT -defined Function-like V37() 1 + 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,f) - (x,M) is Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
- (x,M) is Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
- (x,M) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(x,M) (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) (x,M) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(x,M) (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,f) + (- (x,M)) is Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
(f,f) + (- (x,M)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: ((f,f),(- (x,M))) is set
(f,f) - (x,M) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: ((f,f),(x,M)) is set
- (x,M) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f,f) + (- (x,M)) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: ((f,f),(- (x,M))) is set
f - M is complex real ext-real Element of REAL
((f - x),(f - M)) is non empty Relation-like NAT -defined Function-like V37() 2 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (2)
<*(f - x)*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,(f - x)] is set
{[1,(f - x)]} is non empty trivial Function-like 1 -element set
<*(f - M)*> is non empty trivial Relation-like NAT -defined Function-like one-to-one constant V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V153() decreasing non-decreasing non-increasing set
[1,(f - M)] is set
{[1,(f - M)]} is non empty trivial Function-like 1 -element set
K230(<*(f - x)*>,<*(f - M)*>) is non empty Relation-like NAT -defined Function-like V37() 1 + 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((x,M)) is complex real ext-real Element of REAL
(x,M) . 1 is complex real ext-real Element of REAL
((x,M)) is complex real ext-real Element of REAL
(x,M) . 2 is complex real ext-real Element of REAL
((f,f)) is complex real ext-real Element of REAL
(f,f) . 1 is complex real ext-real Element of REAL
((f,f)) is complex real ext-real Element of REAL
(f,f) . 2 is complex real ext-real Element of REAL
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
bool the carrier of (f) is set
(f) is non empty strict Reflexive discerning V86() triangle Discerning MetrStruct
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
(f) is Relation-like [:(f),(f):] -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:[:(f),(f):],REAL:]
[:(f),(f):] is set
[:[:(f),(f):],REAL:] is complex-valued ext-real-valued real-valued set
bool [:[:(f),(f):],REAL:] is set
MetrStruct(# (f),(f) #) is strict MetrStruct
the carrier of (f) is non empty set
bool the carrier of (f) is set
f is Element of bool the carrier of (f)
(f) | f is strict SubSpace of (f)
x is non empty Element of bool the carrier of (f)
(f) | x is non empty strict Reflexive discerning V86() triangle Discerning SubSpace of (f)
TopSpaceMetr ((f) | x) is non empty strict TopSpace-like TopStruct
the carrier of ((f) | x) is non empty set
Family_open_set ((f) | x) is Element of bool (bool the carrier of ((f) | x))
bool the carrier of ((f) | x) is set
bool (bool the carrier of ((f) | x)) is set
TopStruct(# the carrier of ((f) | x),(Family_open_set ((f) | x)) #) is strict TopStruct
the topology of (f) is Element of bool (bool the carrier of (f))
bool (bool the carrier of (f)) is set
TopStruct(# the carrier of (f), the topology of (f) #) is strict TopStruct
TopSpaceMetr (f) is non empty strict TopSpace-like TopStruct
Family_open_set (f) is Element of bool (bool the carrier of (f))
bool (bool the carrier of (f)) is set
TopStruct(# the carrier of (f),(Family_open_set (f)) #) is strict TopStruct
M is SubSpace of (f)
[#] M is non proper Element of bool the carrier of M
the carrier of M is set
bool the carrier of M is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
M is Relation-like Function-like complex-valued ext-real-valued real-valued set
x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
M is Relation-like Function-like complex-valued ext-real-valued real-valued set
f + x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f + x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (f,x) is set
M + M is Relation-like Function-like complex-valued ext-real-valued real-valued set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
M is Relation-like Function-like complex-valued ext-real-valued real-valued set
f is complex real ext-real set
f * x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the Mult of (f) is Relation-like [:REAL, the carrier of (f):] -defined the carrier of (f) -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of (f):], the carrier of (f):]
[:REAL, the carrier of (f):] is V37() set
[:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
bool [:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
the Mult of (f) . (f,x) is set
f * x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f multreal is Relation-like REAL -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
multreal [;] (f,K83(REAL)) is set
x (#) (f multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) M is Relation-like Function-like complex-valued ext-real-valued real-valued set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
Seg f is V37() f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= f ) } is set
(Seg f) --> 0 is Relation-like Seg f -defined RAT -valued INT -valued {0} -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of bool [:(Seg f),{0}:]
[:(Seg f),{0}:] is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg f),{0}:] is set
0. (f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like zero complex-valued ext-real-valued real-valued Element of the carrier of (f)
the ZeroF of (f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
the addF of (f) is Relation-like [: the carrier of (f), the carrier of (f):] -defined the carrier of (f) -valued Function-like quasi_total Element of bool [:[: the carrier of (f), the carrier of (f):], the carrier of (f):]
[: the carrier of (f), the carrier of (f):] is set
[:[: the carrier of (f), the carrier of (f):], the carrier of (f):] is set
bool [:[: the carrier of (f), the carrier of (f):], the carrier of (f):] is set
the Mult of (f) is Relation-like [:REAL, the carrier of (f):] -defined the carrier of (f) -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of (f):], the carrier of (f):]
[:REAL, the carrier of (f):] is V37() set
[:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
bool [:[:REAL, the carrier of (f):], the carrier of (f):] is V37() set
RLSStruct(# the carrier of (f), the ZeroF of (f), the addF of (f), the Mult of (f) #) is strict RLSStruct
RealVectSpace (Seg f) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
Funcs ((Seg f),REAL) is non empty FUNCTION_DOMAIN of Seg f, REAL
RealFuncZero (Seg f) is Relation-like Seg f -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of Funcs ((Seg f),REAL)
(Seg f) --> 0 is Relation-like Seg f -defined NAT -valued RAT -valued INT -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of bool [:(Seg f),NAT:]
[:(Seg f),NAT:] is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg f),NAT:] is set
RealFuncAdd (Seg f) is Relation-like [:(Funcs ((Seg f),REAL)),(Funcs ((Seg f),REAL)):] -defined Funcs ((Seg f),REAL) -valued Function-like quasi_total Element of bool [:[:(Funcs ((Seg f),REAL)),(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):]
[:(Funcs ((Seg f),REAL)),(Funcs ((Seg f),REAL)):] is set
[:[:(Funcs ((Seg f),REAL)),(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):] is set
bool [:[:(Funcs ((Seg f),REAL)),(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):] is set
RealFuncExtMult (Seg f) is Relation-like [:REAL,(Funcs ((Seg f),REAL)):] -defined Funcs ((Seg f),REAL) -valued Function-like quasi_total Element of bool [:[:REAL,(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):]
[:REAL,(Funcs ((Seg f),REAL)):] is V37() set
[:[:REAL,(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):] is V37() set
bool [:[:REAL,(Funcs ((Seg f),REAL)):],(Funcs ((Seg f),REAL)):] is V37() set
RLSStruct(# (Funcs ((Seg f),REAL)),(RealFuncZero (Seg f)),(RealFuncAdd (Seg f)),(RealFuncExtMult (Seg f)) #) is strict RLSStruct
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty strict Reflexive discerning V86() triangle Discerning MetrStruct
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
(f) is Relation-like [:(f),(f):] -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:[:(f),(f):],REAL:]
[:(f),(f):] is set
[:[:(f),(f):],REAL:] is complex-valued ext-real-valued real-valued set
bool [:[:(f),(f):],REAL:] is set
MetrStruct(# (f),(f) #) is strict MetrStruct
the carrier of (f) is non empty set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
TopSpaceMetr (f) is non empty strict TopSpace-like TopStruct
Family_open_set (f) is Element of bool (bool the carrier of (f))
bool the carrier of (f) is set
bool (bool the carrier of (f)) is set
TopStruct(# the carrier of (f),(Family_open_set (f)) #) is strict TopStruct
the carrier of (TopSpaceMetr (f)) is non empty set
the topology of (f) is Element of bool (bool the carrier of (f))
bool the carrier of (f) is set
bool (bool the carrier of (f)) is set
TopStruct(# the carrier of (f), the topology of (f) #) is strict TopStruct
the carrier of TopStruct(# the carrier of (f), the topology of (f) #) is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
x is Relation-like Function-like complex-valued ext-real-valued real-valued set
- f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
- x is Relation-like Function-like complex-valued ext-real-valued real-valued set
(- 1) (#) x is Relation-like Function-like complex-valued ext-real-valued real-valued set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
M is Relation-like Function-like complex-valued ext-real-valued real-valued set
x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
M is Relation-like Function-like complex-valued ext-real-valued real-valued set
f - x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- x is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
x (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- x) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
f + (- x) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
addreal .: (f,(- x)) is set
f - x is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
diffreal .: (f,x) is set
- x is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f + (- x) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
addreal .: (f,(- x)) is set
M - M is Relation-like Function-like complex-valued ext-real-valued real-valued set
- M is Relation-like Function-like complex-valued ext-real-valued real-valued set
(- 1) (#) M is Relation-like Function-like complex-valued ext-real-valued real-valued set
M + (- M) is Relation-like Function-like complex-valued ext-real-valued real-valued set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
0. (f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like zero complex-valued ext-real-valued real-valued Element of the carrier of (f)
the carrier of (f) is non empty set
the ZeroF of (f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
(f) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
f |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
Seg f is V37() f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= f ) } is set
(Seg f) --> 0 is Relation-like Seg f -defined RAT -valued INT -valued {0} -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of bool [:(Seg f),{0}:]
[:(Seg f),{0}:] is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg f),{0}:] is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- f is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
- f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f (#) compreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- 1) (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) ((- 1) multreal) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((- f)) is complex real ext-real non negative Element of REAL
sqr (- f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(- f) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(- f) (#) (- f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((- f),(- f)) is set
Sum (sqr (- f)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (- f))) is complex real ext-real Element of REAL
(f) is complex real ext-real non negative Element of REAL
sqr f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f (#) f is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: (f,f) is set
Sum (sqr f) is complex real ext-real Element of REAL
sqrt (Sum (sqr f)) is complex real ext-real Element of REAL
f is ordinal natural complex real ext-real V37() cardinal set
f is set
f -tuples_on f is functional FinSequence-membered FinSequenceSet of f
f * is non empty functional FinSequence-membered FinSequenceSet of f
{ b1 where b1 is Relation-like NAT -defined f -valued Function-like V37() FinSequence-like FinSubsequence-like Element of f * : len b1 = f } is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is non empty functional FinSequence-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
f is set
x is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
f is ordinal natural complex real ext-real V37() cardinal set
f |-> 1 is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
Seg f is V37() f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= f ) } is set
(Seg f) --> 1 is Relation-like Seg f -defined RAT -valued INT -valued {1} -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of bool [:(Seg f),{1}:]
[:(Seg f),{1}:] is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg f),{1}:] is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f |-> 1 is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
Seg f is V37() f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= f ) } is set
(Seg f) --> 1 is Relation-like Seg f -defined RAT -valued INT -valued {1} -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of bool [:(Seg f),{1}:]
[:(Seg f),{1}:] is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg f),{1}:] is set
(f) is non empty functional FinSequence-membered real-functions-membered FinSequenceSet of REAL
f is ordinal natural complex real ext-real V37() cardinal set
(f) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
(f) is non empty functional FinSequence-membered real-functions-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
f |-> 1 is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
Seg f is V37() f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= f ) } is set
(Seg f) --> 1 is Relation-like Seg f -defined RAT -valued INT -valued {1} -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of bool [:(Seg f),{1}:]
[:(Seg f),{1}:] is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg f),{1}:] is set
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
(f) is non empty functional FinSequence-membered real-functions-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
f |-> 1 is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
Seg f is V37() f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= f ) } is set
(Seg f) --> 1 is Relation-like Seg f -defined RAT -valued INT -valued {1} -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of bool [:(Seg f),{1}:]
[:(Seg f),{1}:] is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg f),{1}:] is set
(f,(f)) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
(f) (#) () is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
() . 1 is complex real ext-real Element of REAL
f |-> (() . 1) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
(Seg f) --> (() . 1) is Relation-like Seg f -defined {(() . 1)} -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:(Seg f),{(() . 1)}:]
{(() . 1)} is non empty trivial 1 -element V159() V160() V161() set
[:(Seg f),{(() . 1)}:] is complex-valued ext-real-valued real-valued set
bool [:(Seg f),{(() . 1)}:] is set
abs 1 is complex real ext-real V34() Element of REAL
f |-> (abs 1) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
(Seg f) --> (abs 1) is Relation-like Seg f -defined RAT -valued {(abs 1)} -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:(Seg f),{(abs 1)}:]
{(abs 1)} is non empty trivial 1 -element V159() V160() V161() V162() set
[:(Seg f),{(abs 1)}:] is RAT -valued complex-valued ext-real-valued real-valued set
bool [:(Seg f),{(abs 1)}:] is set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
(f) is non empty functional FinSequence-membered real-functions-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
f |-> 1 is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
Seg f is V37() f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= f ) } is set
(Seg f) --> 1 is Relation-like Seg f -defined RAT -valued INT -valued {1} -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of bool [:(Seg f),{1}:]
[:(Seg f),{1}:] is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg f),{1}:] is set
((f)) is complex real ext-real non negative Element of REAL
sqr (f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f) (#) (f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f),(f)) is set
Sum (sqr (f)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f))) is complex real ext-real Element of REAL
sqrt f is complex real ext-real set
1 ^2 is complex real ext-real Element of REAL
1 * 1 is complex real ext-real non negative set
f |-> (1 ^2) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
(Seg f) --> (1 ^2) is Relation-like Seg f -defined {(1 ^2)} -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:(Seg f),{(1 ^2)}:]
{(1 ^2)} is non empty trivial 1 -element V159() V160() V161() set
[:(Seg f),{(1 ^2)}:] is complex-valued ext-real-valued real-valued set
bool [:(Seg f),{(1 ^2)}:] is set
f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum f is complex real ext-real Element of REAL
sqrt (Sum f) is complex real ext-real Element of REAL
f * 1 is complex real ext-real Element of REAL
sqrt (f * 1) is complex real ext-real Element of REAL
f is ordinal natural complex real ext-real V37() cardinal set
(f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
(f) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
(f) is non empty functional FinSequence-membered real-functions-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
f |-> 1 is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
Seg f is V37() f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= f ) } is set
(Seg f) --> 1 is Relation-like Seg f -defined RAT -valued INT -valued {1} -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of bool [:(Seg f),{1}:]
[:(Seg f),{1}:] is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg f),{1}:] is set
((f)) is complex real ext-real non negative Element of REAL
sqr (f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f) (#) (f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f),(f)) is set
Sum (sqr (f)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f))) is complex real ext-real Element of REAL
sqrt f is complex real ext-real set
f is ordinal natural complex real ext-real V37() cardinal set
(f) is Relation-like NAT -defined Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (f)
(f) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of (f) is non empty set
(f) is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (f)
(f) is non empty functional FinSequence-membered real-functions-membered FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
f |-> 1 is Relation-like NAT -defined REAL -valued Function-like V37() f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of f -tuples_on REAL
Seg f is V37() f -element V159() V160() V161() V162() V163() V164() Element of bool NAT
{ b1 where b1 is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT : ( 1 <= b1 & b1 <= f ) } is set
(Seg f) --> 1 is Relation-like Seg f -defined RAT -valued INT -valued {1} -valued Function-like quasi_total V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of bool [:(Seg f),{1}:]
[:(Seg f),{1}:] is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg f),{1}:] is set
((f)) is complex real ext-real non negative Element of REAL
sqr (f) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f) (#) sqrreal is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(f) (#) (f) is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
multreal .: ((f),(f)) is set
Sum (sqr (f)) is complex real ext-real Element of REAL
sqrt (Sum (sqr (f))) is complex real ext-real Element of REAL
sqrt f is complex real ext-real set
f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len f is ordinal natural complex real ext-real V33() V34() V37() cardinal V159() V160() V161() V162() V163() V164() Element of NAT
((len f)) is non empty functional FinSequence-membered real-functions-membered FinSequenceSet of REAL
(len f) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = len f } is set
((len f)) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
the carrier of ((len f)) is non empty set
((len f)) is non empty strict Reflexive discerning V86() triangle Discerning MetrStruct
((len f)) is Relation-like [:((len f)),((len f)):] -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:[:((len f)),((len f)):],REAL:]
[:((len f)),((len f)):] is set
[:[:((len f)),((len f)):],REAL:] is complex-valued ext-real-valued real-valued set
bool [:[:((len f)),((len f)):],REAL:] is set
MetrStruct(# ((len f)),((len f)) #) is strict MetrStruct
the carrier of ((len f)) is non empty set
(0) is non empty functional FinSequence-membered real-functions-membered FinSequenceSet of REAL
0 -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = 0 } is set
(0) is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RLTopStruct
0. (0) is empty ordinal natural complex real ext-real non positive non negative Relation-like NAT -defined RAT -valued Function-like one-to-one constant functional V37() cardinal 0 -element FinSequence-like FinSubsequence-like FinSequence-membered zero complex-valued ext-real-valued real-valued natural-valued V159() V160() V161() V162() V163() V164() V165() Element of the carrier of (0)
the carrier of (0) is non empty set
the ZeroF of (0) is empty ordinal natural complex real ext-real non positive non negative Relation-like NAT -defined RAT -valued Function-like one-to-one constant functional V37() cardinal 0 -element FinSequence-like FinSubsequence-like FinSequence-membered complex-valued ext-real-valued real-valued natural-valued V159() V160() V161() V162() V163() V164() V165() Element of the carrier of (0)
{(0. (0))} is non empty trivial functional 1 -element V159() V160() V161() V162() V163() V164() set
<*> REAL is empty ordinal natural complex real ext-real non positive non negative Relation-like NAT -defined REAL -valued Function-like one-to-one constant functional V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered complex-valued ext-real-valued real-valued natural-valued V159() V160() V161() V162() V163() V164() V165() FinSequence of REAL
{(<*> REAL)} is non empty trivial functional 1 -element V159() V160() V161() V162() V163() V164() set