:: TOPREALC semantic presentation

REAL is non empty non trivial non finite complex-membered ext-real-membered real-membered V132() set
NAT is non trivial ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V132() Element of bool REAL
bool REAL is non trivial non finite set
omega is non trivial ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V132() set
bool omega is non trivial non finite set
bool NAT is non trivial non finite set
COMPLEX is non empty non trivial non finite complex-membered V132() set
RAT is non empty non trivial non finite complex-membered ext-real-membered real-membered rational-membered V132() set
INT is non empty non trivial non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V132() set
[:REAL,REAL:] is Relation-like non trivial complex-valued ext-real-valued real-valued non finite set
bool [:REAL,REAL:] is non trivial non finite set
K282() is non empty strict multMagma
the carrier of K282() is non empty set
K287() is non empty strict V107() V108() V109() V111() V148() V149() V150() V151() V152() V153() multMagma
K288() is non empty strict V109() V111() V151() V152() V153() M13(K287())
K289() is non empty strict V107() V109() V111() V151() V152() V153() V154() M16(K287(),K288())
K291() is non empty strict V107() V109() V111() multMagma
K292() is non empty strict V107() V109() V111() V154() M13(K291())
1 is non empty ext-real positive non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
[:1,1:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:1,1:] is finite V49() set
[:[:1,1:],1:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:[:1,1:],1:] is finite V49() set
[:[:1,1:],REAL:] is Relation-like complex-valued ext-real-valued real-valued set
bool [:[:1,1:],REAL:] is set
[:[:REAL,REAL:],REAL:] is Relation-like non trivial complex-valued ext-real-valued real-valued non finite set
bool [:[:REAL,REAL:],REAL:] is non trivial non finite set
2 is non empty ext-real positive non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
[:2,2:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
[:[:2,2:],REAL:] is Relation-like complex-valued ext-real-valued real-valued set
bool [:[:2,2:],REAL:] is set
TOP-REAL 2 is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict L16()
the carrier of (TOP-REAL 2) is non empty set
K596() is non empty strict TopSpace-like real-membered TopStruct
the carrier of K596() is non empty complex-membered ext-real-membered real-membered set
RealSpace is strict real-membered MetrStruct
real_dist is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
MetrStruct(# REAL,real_dist #) is strict MetrStruct
R^1 is non empty strict TopSpace-like real-membered TopStruct
TopSpaceMetr RealSpace is TopStruct
the carrier of RealSpace is complex-membered ext-real-membered real-membered set
Family_open_set RealSpace is Element of bool (bool the carrier of RealSpace)
bool the carrier of RealSpace is set
bool (bool the carrier of RealSpace) is set
TopStruct(# the carrier of RealSpace,(Family_open_set RealSpace) #) is strict TopStruct
K603() is non empty strict TopSpace-like real-membered SubSpace of R^1
[:K603(),K603():] is non empty strict TopSpace-like TopStruct
the carrier of [:K603(),K603():] is non empty set
K605() is non empty strict TopSpace-like real-membered V244() SubSpace of R^1
the carrier of K605() is non empty complex-membered ext-real-membered real-membered set
bool the carrier of K605() is set
bool (bool the carrier of K605()) is set
Tunit_circle 2 is non empty TopSpace-like V245() SubSpace of TOP-REAL 2
0. (TOP-REAL 2) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite 2 -element FinSequence-like FinSubsequence-like zero V268() Element of the carrier of (TOP-REAL 2)
the ZeroF of (TOP-REAL 2) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite 2 -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL 2)
Tcircle ((0. (TOP-REAL 2)),1) is non empty strict TopSpace-like V245() SubSpace of TOP-REAL 2
Sphere ((0. (TOP-REAL 2)),1) is non empty closed V215( TOP-REAL 2) Element of bool the carrier of (TOP-REAL 2)
bool the carrier of (TOP-REAL 2) is set
(TOP-REAL 2) | (Sphere ((0. (TOP-REAL 2)),1)) is non empty strict TopSpace-like SubSpace of TOP-REAL 2
the carrier of (Tunit_circle 2) is non empty set
[: the carrier of K605(), the carrier of (Tunit_circle 2):] is Relation-like set
bool [: the carrier of K605(), the carrier of (Tunit_circle 2):] is set
CircleMap is Relation-like the carrier of K605() -defined the carrier of K605() -defined the carrier of (Tunit_circle 2) -valued the carrier of (Tunit_circle 2) -valued Function-like non empty total total V18( the carrier of K605(), the carrier of (Tunit_circle 2)) V18( the carrier of K605(), the carrier of (Tunit_circle 2)) V19( the carrier of (Tunit_circle 2)) continuous Element of bool [: the carrier of K605(), the carrier of (Tunit_circle 2):]
c[10] is Element of the carrier of (Tunit_circle 2)
0 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative ordinal natural complex complex-valued ext-real-valued real-valued natural-valued real V43() V44() finite finite-yielding V49() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V132() Function-yielding V156() V157() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued V267() V268() Element of NAT
{} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative ordinal natural complex complex-valued ext-real-valued real-valued natural-valued real finite finite-yielding V49() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V132() Function-yielding V156() V157() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued V267() V268() set
the Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative ordinal natural complex complex-valued ext-real-valued real-valued natural-valued real finite finite-yielding V49() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V132() Function-yielding V156() V157() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued V267() V268() set is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative ordinal natural complex complex-valued ext-real-valued real-valued natural-valued real finite finite-yielding V49() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V132() Function-yielding V156() V157() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued V267() V268() set
|[1,0]| is Relation-like NAT -defined NAT -defined REAL -valued Function-like non empty complex-valued ext-real-valued real-valued finite 2 -element 2 -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL 2)
<*1*> is Relation-like NAT -defined Function-like one-to-one constant non empty trivial complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing finite 1 -element FinSequence-like FinSubsequence-like V268() set
[1,1] is set
{1,1} is non empty finite V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
{1} is non empty trivial finite V49() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
{{1,1},{1}} is non empty finite V49() set
{[1,1]} is Relation-like Function-like constant non empty trivial finite 1 -element V268() set
<*0*> is Relation-like NAT -defined Function-like one-to-one constant non empty trivial complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing finite 1 -element FinSequence-like FinSubsequence-like Function-yielding V156() V157() V267() V268() set
[1,0] is set
{1,0} is non empty finite V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
{{1,0},{1}} is non empty finite V49() set
{[1,0]} is Relation-like Function-like constant non empty trivial finite 1 -element V268() set
<*1*> ^ <*0*> is Relation-like NAT -defined NAT -defined Function-like non empty complex-valued ext-real-valued real-valued finite K107(1,1) -element FinSequence-like FinSubsequence-like V268() set
K107(1,1) is non empty ext-real positive non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
Topen_unit_circle c[10] is non empty strict TopSpace-like SubSpace of Tunit_circle 2
the carrier of (Topen_unit_circle c[10]) is non empty set
].0,1.[ is non empty complex-membered ext-real-membered real-membered open Element of bool REAL
R^1 ].0,1.[ is non empty complex-membered ext-real-membered real-membered open Element of bool the carrier of K605()
K605() | (R^1 ].0,1.[) is non empty strict TopSpace-like real-membered SubSpace of K605()
the carrier of (K605() | (R^1 ].0,1.[)) is non empty complex-membered ext-real-membered real-membered set
[: the carrier of (Topen_unit_circle c[10]), the carrier of (K605() | (R^1 ].0,1.[)):] is Relation-like complex-valued ext-real-valued real-valued set
bool [: the carrier of (Topen_unit_circle c[10]), the carrier of (K605() | (R^1 ].0,1.[)):] is set
c[-10] is Element of the carrier of (Tunit_circle 2)
K504(1) is non empty ext-real non positive negative complex real Element of REAL
|[K504(1),0]| is Relation-like NAT -defined NAT -defined REAL -valued Function-like non empty complex-valued ext-real-valued real-valued finite 2 -element 2 -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL 2)
<*K504(1)*> is Relation-like NAT -defined Function-like one-to-one constant non empty trivial complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing finite 1 -element FinSequence-like FinSubsequence-like V268() set
[1,K504(1)] is set
{1,K504(1)} is non empty finite complex-membered ext-real-membered real-membered set
{{1,K504(1)},{1}} is non empty finite V49() set
{[1,K504(1)]} is Relation-like Function-like constant non empty trivial finite 1 -element V268() set
<*K504(1)*> ^ <*0*> is Relation-like NAT -defined NAT -defined Function-like non empty complex-valued ext-real-valued real-valued finite K107(1,1) -element FinSequence-like FinSubsequence-like V268() set
Topen_unit_circle c[-10] is non empty strict TopSpace-like SubSpace of Tunit_circle 2
the carrier of (Topen_unit_circle c[-10]) is non empty set
K513(1,2) is non empty ext-real positive non negative complex real Element of REAL
2 " is non empty ext-real positive non negative complex real set
1 * (2 ") is non empty ext-real positive non negative complex real set
3 is non empty ext-real positive non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
K513(3,2) is non empty ext-real positive non negative complex real Element of REAL
3 * (2 ") is non empty ext-real positive non negative complex real set
].K513(1,2),K513(3,2).[ is non empty complex-membered ext-real-membered real-membered open Element of bool REAL
R^1 ].K513(1,2),K513(3,2).[ is non empty complex-membered ext-real-membered real-membered open Element of bool the carrier of K605()
K605() | (R^1 ].K513(1,2),K513(3,2).[) is non empty strict TopSpace-like real-membered SubSpace of K605()
the carrier of (K605() | (R^1 ].K513(1,2),K513(3,2).[)) is non empty complex-membered ext-real-membered real-membered set
[: the carrier of (Topen_unit_circle c[-10]), the carrier of (K605() | (R^1 ].K513(1,2),K513(3,2).[)):] is Relation-like complex-valued ext-real-valued real-valued set
bool [: the carrier of (Topen_unit_circle c[-10]), the carrier of (K605() | (R^1 ].K513(1,2),K513(3,2).[)):] is set
K759() is set
{{}} is functional non empty trivial finite V49() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered complex-functions-membered ext-real-functions-membered real-functions-membered set
{{}} * is functional non empty FinSequence-membered FinSequenceSet of {{}}
[:({{}} *),{{}}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:({{}} *),{{}}:] is set
PFuncs (({{}} *),{{}}) is functional complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered set
[:COMPLEX,COMPLEX:] is Relation-like non trivial complex-valued non finite set
bool [:COMPLEX,COMPLEX:] is non trivial non finite set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is Relation-like non trivial complex-valued non finite set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non trivial non finite set
[:RAT,RAT:] is Relation-like RAT -valued non trivial complex-valued ext-real-valued real-valued non finite set
bool [:RAT,RAT:] is non trivial non finite set
[:[:RAT,RAT:],RAT:] is Relation-like RAT -valued non trivial complex-valued ext-real-valued real-valued non finite set
bool [:[:RAT,RAT:],RAT:] is non trivial non finite set
[:INT,INT:] is Relation-like RAT -valued INT -valued non trivial complex-valued ext-real-valued real-valued non finite set
bool [:INT,INT:] is non trivial non finite set
[:[:INT,INT:],INT:] is Relation-like RAT -valued INT -valued non trivial complex-valued ext-real-valued real-valued non finite set
bool [:[:INT,INT:],INT:] is non trivial non finite set
[:NAT,NAT:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
[:[:NAT,NAT:],NAT:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:[:NAT,NAT:],NAT:] is set
{{},1} is non empty finite V49() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
Euclid 0 is non empty V138() V139() strict Reflexive discerning V193() triangle MetrStruct
REAL 0 is functional non empty FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
0 -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
REAL * is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V268() Element of REAL * : len b₁ = 0 } is set
Pitag_dist 0 is Relation-like [:(REAL 0),(REAL 0):] -defined REAL -valued Function-like total V18([:(REAL 0),(REAL 0):], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:(REAL 0),(REAL 0):],REAL:]
[:(REAL 0),(REAL 0):] is Relation-like set
[:[:(REAL 0),(REAL 0):],REAL:] is Relation-like complex-valued ext-real-valued real-valued set
bool [:[:(REAL 0),(REAL 0):],REAL:] is set
MetrStruct(# (REAL 0),(Pitag_dist 0) #) is strict MetrStruct
TopSpaceMetr (Euclid 0) is trivial finite TopStruct
the carrier of (Euclid 0) is non empty set
Family_open_set (Euclid 0) is Element of bool (bool the carrier of (Euclid 0))
bool the carrier of (Euclid 0) is set
bool (bool the carrier of (Euclid 0)) is set
TopStruct(# the carrier of (Euclid 0),(Family_open_set (Euclid 0)) #) is non empty strict TopStruct
TOP-REAL 0 is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict L16()
0. (TOP-REAL 0) is Relation-like non-empty empty-yielding NAT -defined REAL -valued Function-like one-to-one constant functional empty ext-real non positive non negative ordinal natural complex complex-valued ext-real-valued real-valued natural-valued real finite finite-yielding V49() cardinal 0 -element FinSequence-like FinSubsequence-like FinSequence-membered zero complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V132() Function-yielding V156() V157() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued V267() V268() Element of the carrier of (TOP-REAL 0)
the carrier of (TOP-REAL 0) is non empty set
the ZeroF of (TOP-REAL 0) is Relation-like non-empty empty-yielding NAT -defined REAL -valued Function-like one-to-one constant functional empty ext-real non positive non negative ordinal natural complex complex-valued ext-real-valued real-valued natural-valued real finite finite-yielding V49() cardinal 0 -element FinSequence-like FinSubsequence-like FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V132() Function-yielding V156() V157() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued V267() V268() Element of the carrier of (TOP-REAL 0)
{(0. (TOP-REAL 0))} is functional non empty trivial finite V49() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered complex-functions-membered ext-real-functions-membered real-functions-membered set
the carrier of R^1 is non empty complex-membered ext-real-membered real-membered set
sqrt 0 is ext-real non negative complex real Element of REAL
<*> REAL is Relation-like non-empty empty-yielding NAT -defined REAL -valued Function-like one-to-one constant functional empty ext-real non positive non negative ordinal natural complex complex-valued ext-real-valued real-valued natural-valued real finite finite-yielding V49() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V132() Function-yielding V156() V157() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued V267() V268() FinSequence of REAL
Sum (<*> REAL) is ext-real complex real Element of REAL
m is trivial finite set
r is set
card m is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of omega
card r is ordinal cardinal set
m is ext-real complex real set
m ^2 is ext-real complex real set
m * m is ext-real non negative complex real set
m is non empty ext-real positive non negative complex real set
m ^2 is ext-real non negative complex real set
m * m is non empty ext-real positive non negative complex real set
m is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative ordinal natural complex complex-valued ext-real-valued real-valued natural-valued real finite finite-yielding V49() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V132() Function-yielding V156() V157() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued V267() V268() set
sqr m is Relation-like non-empty empty-yielding NAT -defined REAL -valued RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative ordinal natural complex complex-valued ext-real-valued real-valued natural-valued real finite finite-yielding V49() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V132() Function-yielding V156() V157() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued V267() V268() FinSequence of REAL
m (#) m is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative ordinal natural complex complex-valued ext-real-valued real-valued natural-valued real finite finite-yielding V49() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V132() Function-yielding V156() V157() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued V267() V268() set
K857() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K857() .: (m,m) is Relation-like Function-like set
<:m,m:> is Relation-like Function-like set
<:m,m:> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
sqrreal is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
m (#) sqrreal is Relation-like non-empty empty-yielding NAT -defined REAL -valued RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative ordinal natural complex complex-valued ext-real-valued real-valued natural-valued real finite finite-yielding V49() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V132() Function-yielding V156() V157() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued V267() V268() set
|.m.| is ext-real non negative complex real Element of REAL
Sum (sqr m) is ext-real complex real Element of REAL
sqrt (Sum (sqr m)) is ext-real complex real Element of REAL
m is complex set
r is complex set
m + r is complex set
T is Relation-like Function-like complex-valued set
(m + r) (#) T is Relation-like Function-like complex-valued set
m (#) T is Relation-like Function-like complex-valued set
r (#) T is Relation-like Function-like complex-valued set
(m (#) T) + (r (#) T) is Relation-like Function-like complex-valued set
dom ((m + r) (#) T) is set
dom T is set
dom (m (#) T) is set
dom (r (#) T) is set
dom ((m (#) T) + (r (#) T)) is set
(dom (m (#) T)) /\ (dom (r (#) T)) is set
f is set
((m + r) (#) T) . f is complex set
((m (#) T) + (r (#) T)) . f is complex set
T . f is complex set
(T . f) * (m + r) is complex set
(T . f) * m is complex set
(T . f) * r is complex set
((T . f) * m) + ((T . f) * r) is complex set
(m (#) T) . f is complex set
((m (#) T) . f) + ((T . f) * r) is complex set
(r (#) T) . f is complex set
((m (#) T) . f) + ((r (#) T) . f) is complex set
m is complex set
r is complex set
m - r is complex set
- r is complex set
m + (- r) is complex set
T is Relation-like Function-like complex-valued set
(m - r) (#) T is Relation-like Function-like complex-valued set
m (#) T is Relation-like Function-like complex-valued set
r (#) T is Relation-like Function-like complex-valued set
(m (#) T) - (r (#) T) is Relation-like Function-like complex-valued set
- (r (#) T) is Relation-like Function-like complex-valued set
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) (r (#) T) is Relation-like Function-like complex-valued set
(m (#) T) + (- (r (#) T)) is Relation-like Function-like complex-valued set
(- r) (#) T is Relation-like Function-like complex-valued set
(m (#) T) + ((- r) (#) T) is Relation-like Function-like complex-valued set
m is complex set
r is Relation-like Function-like complex-valued set
r (/) m is Relation-like Function-like complex-valued set
1 / m is complex set
m " is complex set
1 * (m ") is complex set
(1 / m) (#) r is Relation-like Function-like complex-valued set
T is Relation-like Function-like complex-valued set
T (/) m is Relation-like Function-like complex-valued set
(1 / m) (#) T is Relation-like Function-like complex-valued set
(r (/) m) + (T (/) m) is Relation-like Function-like complex-valued set
r + T is Relation-like Function-like complex-valued set
(r + T) (/) m is Relation-like Function-like complex-valued set
(1 / m) (#) (r + T) is Relation-like Function-like complex-valued set
dom ((r (/) m) + (T (/) m)) is set
dom (r (/) m) is set
dom (T (/) m) is set
(dom (r (/) m)) /\ (dom (T (/) m)) is set
dom r is set
dom T is set
dom ((r + T) (/) m) is set
dom (r + T) is set
(dom r) /\ (dom T) is set
f is set
((r (/) m) + (T (/) m)) . f is complex set
((r + T) (/) m) . f is complex set
(r + T) . f is complex set
((r + T) . f) / m is complex Element of COMPLEX
((r + T) . f) * (m ") is complex set
r . f is complex set
T . f is complex set
(r . f) + (T . f) is complex set
((r . f) + (T . f)) / m is complex Element of COMPLEX
((r . f) + (T . f)) * (m ") is complex set
(r . f) / m is complex Element of COMPLEX
(r . f) * (m ") is complex set
(T . f) / m is complex Element of COMPLEX
(T . f) * (m ") is complex set
((r . f) / m) + ((T . f) / m) is complex Element of COMPLEX
(T (/) m) . f is complex set
((r . f) / m) + ((T (/) m) . f) is complex set
(r (/) m) . f is complex set
((r (/) m) . f) + ((T (/) m) . f) is complex set
m is complex set
r is Relation-like Function-like complex-valued set
r (/) m is Relation-like Function-like complex-valued set
1 / m is complex set
m " is complex set
1 * (m ") is complex set
(1 / m) (#) r is Relation-like Function-like complex-valued set
T is Relation-like Function-like complex-valued set
T (/) m is Relation-like Function-like complex-valued set
(1 / m) (#) T is Relation-like Function-like complex-valued set
(r (/) m) - (T (/) m) is Relation-like Function-like complex-valued set
- (T (/) m) is Relation-like Function-like complex-valued set
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) (T (/) m) is Relation-like Function-like complex-valued set
(r (/) m) + (- (T (/) m)) is Relation-like Function-like complex-valued set
r - T is Relation-like Function-like complex-valued set
- T is Relation-like Function-like complex-valued set
(- 1) (#) T is Relation-like Function-like complex-valued set
r + (- T) is Relation-like Function-like complex-valued set
(r - T) (/) m is Relation-like Function-like complex-valued set
(1 / m) (#) (r - T) is Relation-like Function-like complex-valued set
(- T) (/) m is Relation-like Function-like complex-valued set
(1 / m) (#) (- T) is Relation-like Function-like complex-valued set
(r (/) m) + ((- T) (/) m) is Relation-like Function-like complex-valued set
m is complex set
r is complex set
m * r is complex set
T is Relation-like Function-like complex-valued set
T (/) m is Relation-like Function-like complex-valued set
1 / m is complex set
m " is complex set
1 * (m ") is complex set
(1 / m) (#) T is Relation-like Function-like complex-valued set
r (#) T is Relation-like Function-like complex-valued set
f is Relation-like Function-like complex-valued set
f (/) r is Relation-like Function-like complex-valued set
1 / r is complex set
r " is complex set
1 * (r ") is complex set
(1 / r) (#) f is Relation-like Function-like complex-valued set
(T (/) m) - (f (/) r) is Relation-like Function-like complex-valued set
- (f (/) r) is Relation-like Function-like complex-valued set
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) (f (/) r) is Relation-like Function-like complex-valued set
(T (/) m) + (- (f (/) r)) is Relation-like Function-like complex-valued set
m (#) f is Relation-like Function-like complex-valued set
(r (#) T) - (m (#) f) is Relation-like Function-like complex-valued set
- (m (#) f) is Relation-like Function-like complex-valued set
(- 1) (#) (m (#) f) is Relation-like Function-like complex-valued set
(r (#) T) + (- (m (#) f)) is Relation-like Function-like complex-valued set
((r (#) T) - (m (#) f)) (/) (m * r) is Relation-like Function-like complex-valued set
1 / (m * r) is complex set
(m * r) " is complex set
1 * ((m * r) ") is complex set
(1 / (m * r)) (#) ((r (#) T) - (m (#) f)) is Relation-like Function-like complex-valued set
dom (T (/) m) is set
dom T is set
dom (f (/) r) is set
dom f is set
dom (r (#) T) is set
dom (m (#) f) is set
dom ((T (/) m) - (f (/) r)) is set
(dom (T (/) m)) /\ (dom (f (/) r)) is set
dom ((r (#) T) - (m (#) f)) is set
(dom (r (#) T)) /\ (dom (m (#) f)) is set
dom (((r (#) T) - (m (#) f)) (/) (m * r)) is set
g is set
((T (/) m) - (f (/) r)) . g is complex set
(((r (#) T) - (m (#) f)) (/) (m * r)) . g is complex set
(T (/) m) . g is complex set
(f (/) r) . g is complex set
((T (/) m) . g) - ((f (/) r) . g) is complex set
- ((f (/) r) . g) is complex set
((T (/) m) . g) + (- ((f (/) r) . g)) is complex set
T . g is complex set
(T . g) / m is complex Element of COMPLEX
(T . g) * (m ") is complex set
((T . g) / m) - ((f (/) r) . g) is complex set
((T . g) / m) + (- ((f (/) r) . g)) is complex set
f . g is complex set
(f . g) / r is complex Element of COMPLEX
(f . g) * (r ") is complex set
((T . g) / m) - ((f . g) / r) is complex Element of COMPLEX
- ((f . g) / r) is complex set
((T . g) / m) + (- ((f . g) / r)) is complex set
(T . g) * r is complex set
(f . g) * m is complex set
((T . g) * r) - ((f . g) * m) is complex set
- ((f . g) * m) is complex set
((T . g) * r) + (- ((f . g) * m)) is complex set
(((T . g) * r) - ((f . g) * m)) / (m * r) is complex Element of COMPLEX
(((T . g) * r) - ((f . g) * m)) * ((m * r) ") is complex set
(m (#) f) . g is complex set
((T . g) * r) - ((m (#) f) . g) is complex set
- ((m (#) f) . g) is complex set
((T . g) * r) + (- ((m (#) f) . g)) is complex set
(((T . g) * r) - ((m (#) f) . g)) / (m * r) is complex Element of COMPLEX
(((T . g) * r) - ((m (#) f) . g)) * ((m * r) ") is complex set
(r (#) T) . g is complex set
((r (#) T) . g) - ((m (#) f) . g) is complex set
((r (#) T) . g) + (- ((m (#) f) . g)) is complex set
(((r (#) T) . g) - ((m (#) f) . g)) / (m * r) is complex Element of COMPLEX
(((r (#) T) . g) - ((m (#) f) . g)) * ((m * r) ") is complex set
((r (#) T) - (m (#) f)) . g is complex set
(((r (#) T) - (m (#) f)) . g) / (m * r) is complex Element of COMPLEX
(((r (#) T) - (m (#) f)) . g) * ((m * r) ") is complex set
m is complex set
r is Relation-like Function-like complex-valued set
r (/) m is Relation-like Function-like complex-valued set
1 / m is complex set
m " is complex set
1 * (m ") is complex set
(1 / m) (#) r is Relation-like Function-like complex-valued set
T is Relation-like Function-like complex-valued set
(r (/) m) - T is Relation-like Function-like complex-valued set
- T is Relation-like Function-like complex-valued set
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) T is Relation-like Function-like complex-valued set
(r (/) m) + (- T) is Relation-like Function-like complex-valued set
m (#) T is Relation-like Function-like complex-valued set
r - (m (#) T) is Relation-like Function-like complex-valued set
- (m (#) T) is Relation-like Function-like complex-valued set
(- 1) (#) (m (#) T) is Relation-like Function-like complex-valued set
r + (- (m (#) T)) is Relation-like Function-like complex-valued set
(r - (m (#) T)) (/) m is Relation-like Function-like complex-valued set
(1 / m) (#) (r - (m (#) T)) is Relation-like Function-like complex-valued set
dom ((r (/) m) - T) is set
dom (r (/) m) is set
dom T is set
(dom (r (/) m)) /\ (dom T) is set
dom r is set
(dom r) /\ (dom T) is set
dom (r - (m (#) T)) is set
dom (m (#) T) is set
(dom r) /\ (dom (m (#) T)) is set
dom ((r - (m (#) T)) (/) m) is set
f is set
((r (/) m) - T) . f is complex set
((r - (m (#) T)) (/) m) . f is complex set
(r (/) m) . f is complex set
T . f is complex set
((r (/) m) . f) - (T . f) is complex set
- (T . f) is complex set
((r (/) m) . f) + (- (T . f)) is complex set
r . f is complex set
(r . f) / m is complex Element of COMPLEX
(r . f) * (m ") is complex set
((r . f) / m) - (T . f) is complex set
((r . f) / m) + (- (T . f)) is complex set
m * (T . f) is complex set
(m * (T . f)) / m is complex Element of COMPLEX
(m * (T . f)) * (m ") is complex set
((r . f) / m) - ((m * (T . f)) / m) is complex Element of COMPLEX
- ((m * (T . f)) / m) is complex set
((r . f) / m) + (- ((m * (T . f)) / m)) is complex set
(m (#) T) . f is complex set
((m (#) T) . f) / m is complex Element of COMPLEX
((m (#) T) . f) * (m ") is complex set
((r . f) / m) - (((m (#) T) . f) / m) is complex Element of COMPLEX
- (((m (#) T) . f) / m) is complex set
((r . f) / m) + (- (((m (#) T) . f) / m)) is complex set
(r . f) - ((m (#) T) . f) is complex set
- ((m (#) T) . f) is complex set
(r . f) + (- ((m (#) T) . f)) is complex set
((r . f) - ((m (#) T) . f)) / m is complex Element of COMPLEX
((r . f) - ((m (#) T) . f)) * (m ") is complex set
(r - (m (#) T)) . f is complex set
((r - (m (#) T)) . f) / m is complex Element of COMPLEX
((r - (m (#) T)) . f) * (m ") is complex set
m is complex set
r is complex set
m - r is complex set
- r is complex set
m + (- r) is complex set
T is Relation-like Function-like complex-valued set
(m - r) (#) T is Relation-like Function-like complex-valued set
m (#) T is Relation-like Function-like complex-valued set
r (#) T is Relation-like Function-like complex-valued set
(m (#) T) - (r (#) T) is Relation-like Function-like complex-valued set
- (r (#) T) is Relation-like Function-like complex-valued set
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) (r (#) T) is Relation-like Function-like complex-valued set
(m (#) T) + (- (r (#) T)) is Relation-like Function-like complex-valued set
dom ((m (#) T) - (r (#) T)) is set
dom (m (#) T) is set
dom (r (#) T) is set
(dom (m (#) T)) /\ (dom (r (#) T)) is set
dom T is set
(dom T) /\ (dom (r (#) T)) is set
(dom T) /\ (dom T) is set
dom ((m - r) (#) T) is set
f is set
((m - r) (#) T) . f is complex set
((m (#) T) - (r (#) T)) . f is complex set
T . f is complex set
(m - r) * (T . f) is complex set
m * (T . f) is complex set
r * (T . f) is complex set
(m * (T . f)) - (r * (T . f)) is complex set
- (r * (T . f)) is complex set
(m * (T . f)) + (- (r * (T . f))) is complex set
(r (#) T) . f is complex set
(m * (T . f)) - ((r (#) T) . f) is complex set
- ((r (#) T) . f) is complex set
(m * (T . f)) + (- ((r (#) T) . f)) is complex set
(m (#) T) . f is complex set
((m (#) T) . f) - ((r (#) T) . f) is complex set
((m (#) T) . f) + (- ((r (#) T) . f)) is complex set
m is Relation-like Function-like complex-valued set
r is Relation-like Function-like complex-valued set
m - r is Relation-like Function-like complex-valued set
- r is Relation-like Function-like complex-valued set
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) r is Relation-like Function-like complex-valued set
m + (- r) is Relation-like Function-like complex-valued set
(m - r) ^2 is Relation-like Function-like complex-valued set
(m - r) (#) (m - r) is Relation-like Function-like complex-valued set
r - m is Relation-like Function-like complex-valued set
- m is Relation-like Function-like complex-valued set
(- 1) (#) m is Relation-like Function-like complex-valued set
r + (- m) is Relation-like Function-like complex-valued set
(r - m) ^2 is Relation-like Function-like complex-valued set
(r - m) (#) (r - m) is Relation-like Function-like complex-valued set
dom (m - r) is set
dom m is set
dom r is set
(dom m) /\ (dom r) is set
dom (r - m) is set
(dom r) /\ (dom m) is set
dom ((m - r) ^2) is set
dom ((r - m) ^2) is set
T is set
((m - r) ^2) . T is complex set
((r - m) ^2) . T is complex set
(m - r) . T is complex set
m . T is complex set
r . T is complex set
(m . T) - (r . T) is complex set
- (r . T) is complex set
(m . T) + (- (r . T)) is complex set
(r - m) . T is complex set
(r . T) - (m . T) is complex set
- (m . T) is complex set
(r . T) + (- (m . T)) is complex set
((m - r) . T) * ((m - r) . T) is complex set
((r - m) . T) * ((r - m) . T) is complex set
m is complex set
m ^2 is complex set
m * m is complex set
r is Relation-like Function-like complex-valued set
r (/) m is Relation-like Function-like complex-valued set
1 / m is complex set
m " is complex set
1 * (m ") is complex set
(1 / m) (#) r is Relation-like Function-like complex-valued set
(r (/) m) ^2 is Relation-like Function-like complex-valued set
(r (/) m) (#) (r (/) m) is Relation-like Function-like complex-valued set
r ^2 is Relation-like Function-like complex-valued set
r (#) r is Relation-like Function-like complex-valued set
(r ^2) (/) (m ^2) is Relation-like Function-like complex-valued set
1 / (m ^2) is complex set
(m ^2) " is complex set
1 * ((m ^2) ") is complex set
(1 / (m ^2)) (#) (r ^2) is Relation-like Function-like complex-valued set
dom ((r (/) m) ^2) is set
dom (r (/) m) is set
dom r is set
dom (r ^2) is set
dom ((r ^2) (/) (m ^2)) is set
T is set
((r (/) m) ^2) . T is complex set
((r ^2) (/) (m ^2)) . T is complex set
(r (/) m) . T is complex set
((r (/) m) . T) ^2 is complex set
((r (/) m) . T) * ((r (/) m) . T) is complex set
r . T is complex set
(r . T) / m is complex Element of COMPLEX
(r . T) * (m ") is complex set
((r . T) / m) ^2 is complex Element of COMPLEX
((r . T) / m) * ((r . T) / m) is complex set
(r . T) ^2 is complex set
(r . T) * (r . T) is complex set
((r . T) ^2) / (m ^2) is complex Element of COMPLEX
((r . T) ^2) * ((m ^2) ") is complex set
(r ^2) . T is complex set
((r ^2) . T) / (m ^2) is complex Element of COMPLEX
((r ^2) . T) * ((m ^2) ") is complex set
m is ext-real non negative ordinal natural complex real finite cardinal set
sqrt m is ext-real non negative complex real set
r is ext-real complex real set
m |-> r is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
Seg m is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
(Seg m) --> r is Relation-like Seg m -defined Seg m -defined {r} -valued Function-like constant total total V18( Seg m,{r}) complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() Element of bool [:(Seg m),{r}:]
{r} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(Seg m),{r}:] is Relation-like complex-valued ext-real-valued real-valued finite set
bool [:(Seg m),{r}:] is finite V49() set
T is ext-real complex real set
m |-> T is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
(Seg m) --> T is Relation-like Seg m -defined Seg m -defined {T} -valued Function-like constant total total V18( Seg m,{T}) complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() Element of bool [:(Seg m),{T}:]
{T} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(Seg m),{T}:] is Relation-like complex-valued ext-real-valued real-valued finite set
bool [:(Seg m),{T}:] is finite V49() set
(m |-> r) - (m |-> T) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
- (m |-> T) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) (m |-> T) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K260((- 1)) is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
id REAL is Relation-like REAL -defined REAL -valued Function-like one-to-one non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued increasing non-decreasing Element of bool [:REAL,REAL:]
K857() [;] ((- 1),(id REAL)) is Relation-like Function-like set
dom (id REAL) is non empty set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined Function-like constant non empty total complex-valued ext-real-valued real-valued V315() V316() set
{(- 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(dom (id REAL)),{(- 1)}:] is Relation-like complex-valued ext-real-valued real-valued set
<:((dom (id REAL)) --> (- 1)),(id REAL):> is Relation-like Function-like set
<:((dom (id REAL)) --> (- 1)),(id REAL):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(m |-> T) (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
K853() is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
(m |-> T) (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
(m |-> r) + (- (m |-> T)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K855() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K855() .: ((m |-> r),(- (m |-> T))) is Relation-like Function-like set
<:(m |-> r),(- (m |-> T)):> is Relation-like Function-like set
<:(m |-> r),(- (m |-> T)):> (#) K855() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
K856() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K342(REAL,K855(),(id REAL),K853()) is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K856() .: ((m |-> r),(m |-> T)) is Relation-like Function-like set
<:(m |-> r),(m |-> T):> is Relation-like Function-like set
<:(m |-> r),(m |-> T):> (#) K856() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
|.((m |-> r) - (m |-> T)).| is ext-real non negative complex real Element of REAL
sqr ((m |-> r) - (m |-> T)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
((m |-> r) - (m |-> T)) (#) ((m |-> r) - (m |-> T)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() V317() set
K857() .: (((m |-> r) - (m |-> T)),((m |-> r) - (m |-> T))) is Relation-like Function-like set
<:((m |-> r) - (m |-> T)),((m |-> r) - (m |-> T)):> is Relation-like Function-like set
<:((m |-> r) - (m |-> T)),((m |-> r) - (m |-> T)):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
((m |-> r) - (m |-> T)) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum (sqr ((m |-> r) - (m |-> T))) is ext-real complex real Element of REAL
sqrt (Sum (sqr ((m |-> r) - (m |-> T)))) is ext-real complex real Element of REAL
r - T is ext-real complex real set
- T is ext-real complex real set
r + (- T) is ext-real complex real set
abs (r - T) is ext-real complex real Element of REAL
(sqrt m) * (abs (r - T)) is ext-real complex real set
m |-> (r - T) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
(Seg m) --> (r - T) is Relation-like Seg m -defined Seg m -defined {(r - T)} -valued Function-like constant total total V18( Seg m,{(r - T)}) complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() Element of bool [:(Seg m),{(r - T)}:]
{(r - T)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(Seg m),{(r - T)}:] is Relation-like complex-valued ext-real-valued real-valued finite set
bool [:(Seg m),{(r - T)}:] is finite V49() set
sqr (m |-> (r - T)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(m |-> (r - T)) (#) (m |-> (r - T)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K857() .: ((m |-> (r - T)),(m |-> (r - T))) is Relation-like Function-like set
<:(m |-> (r - T)),(m |-> (r - T)):> is Relation-like Function-like set
<:(m |-> (r - T)),(m |-> (r - T)):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(m |-> (r - T)) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum (sqr (m |-> (r - T))) is ext-real complex real Element of REAL
sqrt (Sum (sqr (m |-> (r - T)))) is ext-real complex real Element of REAL
(r - T) ^2 is ext-real non negative complex real set
(r - T) * (r - T) is ext-real non negative complex real set
m |-> ((r - T) ^2) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
(Seg m) --> ((r - T) ^2) is Relation-like Seg m -defined Seg m -defined {((r - T) ^2)} -valued Function-like constant total total V18( Seg m,{((r - T) ^2)}) complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() V317() Element of bool [:(Seg m),{((r - T) ^2)}:]
{((r - T) ^2)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(Seg m),{((r - T) ^2)}:] is Relation-like complex-valued ext-real-valued real-valued finite set
bool [:(Seg m),{((r - T) ^2)}:] is finite V49() set
Sum (m |-> ((r - T) ^2)) is ext-real complex real set
sqrt (Sum (m |-> ((r - T) ^2))) is ext-real complex real set
m * ((r - T) ^2) is ext-real non negative complex real set
sqrt (m * ((r - T) ^2)) is ext-real non negative complex real set
sqrt ((r - T) ^2) is ext-real non negative complex real set
(sqrt m) * (sqrt ((r - T) ^2)) is ext-real non negative complex real set
m is Relation-like Function-like complex-valued set
r is set
T is complex set
m +* (r,T) is Relation-like Function-like set
g is set
dom (m +* (r,T)) is set
(m +* (r,T)) . g is set
dom m is set
dom m is set
m . g is complex set
dom m is set
m . g is complex set
dom m is set
m is set
r is ext-real non negative ordinal natural complex real finite cardinal set
0* r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of REAL r
REAL r is functional non empty FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
r -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V268() Element of REAL * : len b₁ = r } is set
r |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of r -tuples_on REAL
Seg r is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= r ) } is set
(Seg r) --> 0 is Relation-like Seg r -defined Seg r -defined RAT -valued INT -valued {0} -valued Function-like constant total total V18( Seg r,{0}) complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued V268() V316() V317() Element of bool [:(Seg r),{0}:]
{0} is functional non empty trivial finite V49() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered complex-functions-membered ext-real-functions-membered real-functions-membered set
[:(Seg r),{0}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg r),{0}:] is finite V49() set
T is complex set
(0* r) +* (m,T) is Relation-like NAT -defined Function-like complex-valued finite r -element FinSequence-like FinSubsequence-like V268() set
((0* r) +* (m,T)) ^2 is Relation-like NAT -defined Function-like complex-valued finite r -element FinSequence-like FinSubsequence-like V268() set
((0* r) +* (m,T)) (#) ((0* r) +* (m,T)) is Relation-like NAT -defined Function-like complex-valued finite r -element FinSequence-like FinSubsequence-like V268() set
T ^2 is complex set
T * T is complex set
(0* r) +* (m,(T ^2)) is Relation-like NAT -defined Function-like complex-valued finite r -element FinSequence-like FinSubsequence-like V268() set
dom ((0* r) +* (m,T)) is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom (0* r) is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom ((0* r) +* (m,(T ^2))) is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom (((0* r) +* (m,T)) ^2) is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom (((0* r) +* (m,T)) ^2) is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
p is set
(((0* r) +* (m,T)) ^2) . p is complex set
((0* r) +* (m,(T ^2))) . p is complex set
((0* r) +* (m,T)) . p is complex set
(((0* r) +* (m,T)) . p) ^2 is complex set
(((0* r) +* (m,T)) . p) * (((0* r) +* (m,T)) . p) is complex set
(r |-> 0) . p is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative ordinal natural complex complex-valued ext-real-valued real-valued natural-valued real finite finite-yielding V49() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V132() Function-yielding V156() V157() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued V267() V268() set
{} . m is Relation-like non-empty empty-yielding NAT -defined RAT -valued INT -valued Function-like one-to-one constant functional empty ext-real non positive non negative ordinal natural complex complex-valued ext-real-valued real-valued natural-valued real V44() finite finite-yielding V49() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V132() Function-yielding V156() V157() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued V267() V268() set
m is set
r is ext-real non negative ordinal natural complex real finite cardinal set
Seg r is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= r ) } is set
0* r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of REAL r
REAL r is functional non empty FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
r -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V268() Element of REAL * : len b₁ = r } is set
r |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of r -tuples_on REAL
(Seg r) --> 0 is Relation-like Seg r -defined Seg r -defined RAT -valued INT -valued {0} -valued Function-like constant total total V18( Seg r,{0}) complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued V268() V316() V317() Element of bool [:(Seg r),{0}:]
{0} is functional non empty trivial finite V49() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered complex-functions-membered ext-real-functions-membered real-functions-membered set
[:(Seg r),{0}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg r),{0}:] is finite V49() set
T is ext-real complex real set
(0* r) +* (m,T) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
|.((0* r) +* (m,T)).| is ext-real non negative complex real Element of REAL
sqr ((0* r) +* (m,T)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
((0* r) +* (m,T)) (#) ((0* r) +* (m,T)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
K857() .: (((0* r) +* (m,T)),((0* r) +* (m,T))) is Relation-like Function-like set
<:((0* r) +* (m,T)),((0* r) +* (m,T)):> is Relation-like Function-like set
<:((0* r) +* (m,T)),((0* r) +* (m,T)):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
((0* r) +* (m,T)) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum (sqr ((0* r) +* (m,T))) is ext-real complex real Element of REAL
sqrt (Sum (sqr ((0* r) +* (m,T)))) is ext-real complex real Element of REAL
abs T is ext-real complex real Element of REAL
T ^2 is ext-real non negative complex real set
T * T is ext-real non negative complex real set
((0* r) +* (m,T)) ^2 is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
(0* r) +* (m,(T ^2)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
Sum (((0* r) +* (m,T)) ^2) is ext-real complex real set
m is set
r is ext-real non negative ordinal natural complex real finite cardinal set
0.REAL r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
TOP-REAL r is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict L16()
the carrier of (TOP-REAL r) is non empty set
r |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of r -tuples_on REAL
r -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V268() Element of REAL * : len b₁ = r } is set
Seg r is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= r ) } is set
(Seg r) --> 0 is Relation-like Seg r -defined Seg r -defined RAT -valued INT -valued {0} -valued Function-like constant total total V18( Seg r,{0}) complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued V268() V316() V317() Element of bool [:(Seg r),{0}:]
{0} is functional non empty trivial finite V49() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered complex-functions-membered ext-real-functions-membered real-functions-membered set
[:(Seg r),{0}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg r),{0}:] is finite V49() set
(0.REAL r) +* (m,0) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
dom ((0.REAL r) +* (m,0)) is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom (0.REAL r) is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
f is set
((0.REAL r) +* (m,0)) . f is ext-real complex real set
(0.REAL r) . f is ext-real complex real set
((0.REAL r) +* (m,0)) . f is ext-real complex real set
(0.REAL r) . f is ext-real complex real set
m is set
r is ext-real non negative ordinal natural complex real finite cardinal set
0.REAL r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
TOP-REAL r is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict L16()
the carrier of (TOP-REAL r) is non empty set
r |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of r -tuples_on REAL
r -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V268() Element of REAL * : len b₁ = r } is set
Seg r is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= r ) } is set
(Seg r) --> 0 is Relation-like Seg r -defined Seg r -defined RAT -valued INT -valued {0} -valued Function-like constant total total V18( Seg r,{0}) complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued V268() V316() V317() Element of bool [:(Seg r),{0}:]
{0} is functional non empty trivial finite V49() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered complex-functions-membered ext-real-functions-membered real-functions-membered set
[:(Seg r),{0}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg r),{0}:] is finite V49() set
T is ext-real complex real set
(0.REAL r) +* (m,T) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
f is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
mlt (f,((0.REAL r) +* (m,T))) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K857() .: (f,((0.REAL r) +* (m,T))) is Relation-like Function-like set
<:f,((0.REAL r) +* (m,T)):> is Relation-like Function-like set
<:f,((0.REAL r) +* (m,T)):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
f . m is ext-real complex real set
(f . m) * T is ext-real complex real set
(0.REAL r) +* (m,((f . m) * T)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
dom f is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom ((0.REAL r) +* (m,T)) is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom (mlt (f,((0.REAL r) +* (m,T)))) is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
(dom f) /\ (dom ((0.REAL r) +* (m,T))) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom ((0.REAL r) +* (m,((f . m) * T))) is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom (0.REAL r) is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
p is set
(mlt (f,((0.REAL r) +* (m,T)))) . p is ext-real complex real set
f . p is ext-real complex real set
((0.REAL r) +* (m,T)) . p is ext-real complex real set
(f . p) * (((0.REAL r) +* (m,T)) . p) is ext-real complex real set
(f . p) * T is ext-real complex real set
((0.REAL r) +* (m,((f . m) * T))) . p is ext-real complex real set
(0.REAL r) . p is ext-real complex real set
(f . p) * ((0.REAL r) . p) is ext-real complex real set
(r |-> 0) . p is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative ordinal natural complex complex-valued ext-real-valued real-valued natural-valued real finite finite-yielding V49() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V132() Function-yielding V156() V157() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued V267() V268() set
(f . p) * ((r |-> 0) . p) is Relation-like non-empty empty-yielding NAT -defined REAL -valued Function-like one-to-one constant functional empty ext-real non positive non negative ordinal natural complex complex-valued ext-real-valued real-valued natural-valued real finite finite-yielding V49() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V132() Function-yielding V156() V157() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued V267() V268() FinSequence of REAL
K260((f . p)) is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
id REAL is Relation-like REAL -defined REAL -valued Function-like one-to-one non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued increasing non-decreasing Element of bool [:REAL,REAL:]
K857() [;] ((f . p),(id REAL)) is Relation-like Function-like set
dom (id REAL) is non empty set
(dom (id REAL)) --> (f . p) is Relation-like dom (id REAL) -defined Function-like constant non empty total complex-valued ext-real-valued real-valued set
{(f . p)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(dom (id REAL)),{(f . p)}:] is Relation-like complex-valued ext-real-valued real-valued set
<:((dom (id REAL)) --> (f . p)),(id REAL):> is Relation-like Function-like set
<:((dom (id REAL)) --> (f . p)),(id REAL):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
((r |-> 0) . p) (#) K260((f . p)) is Relation-like non-empty empty-yielding NAT -defined REAL -valued RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative ordinal natural complex complex-valued ext-real-valued real-valued natural-valued real finite finite-yielding V49() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V132() Function-yielding V156() V157() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued V267() V268() set
((0.REAL r) +* (m,((f . m) * T))) . p is ext-real complex real set
m is set
r is ext-real non negative ordinal natural complex real finite cardinal set
0.REAL r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
TOP-REAL r is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict L16()
the carrier of (TOP-REAL r) is non empty set
r |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of r -tuples_on REAL
r -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V268() Element of REAL * : len b₁ = r } is set
Seg r is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= r ) } is set
(Seg r) --> 0 is Relation-like Seg r -defined Seg r -defined RAT -valued INT -valued {0} -valued Function-like constant total total V18( Seg r,{0}) complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued V268() V316() V317() Element of bool [:(Seg r),{0}:]
{0} is functional non empty trivial finite V49() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered complex-functions-membered ext-real-functions-membered real-functions-membered set
[:(Seg r),{0}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg r),{0}:] is finite V49() set
T is ext-real complex real set
(0.REAL r) +* (m,T) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
f is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
|(f,((0.REAL r) +* (m,T)))| is ext-real complex real Element of REAL
mlt (f,((0.REAL r) +* (m,T))) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K857() .: (f,((0.REAL r) +* (m,T))) is Relation-like Function-like set
<:f,((0.REAL r) +* (m,T)):> is Relation-like Function-like set
<:f,((0.REAL r) +* (m,T)):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
Sum (mlt (f,((0.REAL r) +* (m,T)))) is ext-real complex real Element of REAL
f . m is ext-real complex real set
(f . m) * T is ext-real complex real set
(0.REAL r) +* (m,((f . m) * T)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
dom f is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
Sum (0.REAL r) is ext-real complex real set
m is ext-real non negative ordinal natural complex real finite cardinal set
r is ext-real non negative ordinal natural complex real finite cardinal set
0* r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of REAL r
REAL r is functional non empty FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
r -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V268() Element of REAL * : len b₁ = r } is set
r |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of r -tuples_on REAL
Seg r is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= r ) } is set
(Seg r) --> 0 is Relation-like Seg r -defined Seg r -defined RAT -valued INT -valued {0} -valued Function-like constant total total V18( Seg r,{0}) complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued V268() V316() V317() Element of bool [:(Seg r),{0}:]
{0} is functional non empty trivial finite V49() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered complex-functions-membered ext-real-functions-membered real-functions-membered set
[:(Seg r),{0}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg r),{0}:] is finite V49() set
T is complex set
f is Relation-like NAT -defined Function-like complex-valued finite r -element FinSequence-like FinSubsequence-like V268() set
f +* (m,T) is Relation-like NAT -defined Function-like complex-valued finite r -element FinSequence-like FinSubsequence-like V268() set
(f +* (m,T)) - f is Relation-like NAT -defined Function-like complex-valued finite r -element FinSequence-like FinSubsequence-like V268() set
- f is Relation-like NAT -defined Function-like complex-valued finite r -element FinSequence-like FinSubsequence-like V268() set
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) f is Relation-like NAT -defined Function-like complex-valued finite r -element FinSequence-like FinSubsequence-like V268() set
(f +* (m,T)) + (- f) is Relation-like NAT -defined Function-like complex-valued finite r -element FinSequence-like FinSubsequence-like V268() set
f . m is complex set
T - (f . m) is complex set
- (f . m) is complex set
T + (- (f . m)) is complex set
(0* r) +* (m,(T - (f . m))) is Relation-like NAT -defined Function-like complex-valued finite r -element FinSequence-like FinSubsequence-like V268() set
dom ((f +* (m,T)) - f) is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom (f +* (m,T)) is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom f is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
(dom (f +* (m,T))) /\ (dom f) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom (0* r) is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom ((0* r) +* (m,(T - (f . m)))) is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom ((f +* (m,T)) - f) is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
g is set
((f +* (m,T)) - f) . g is complex set
((0* r) +* (m,(T - (f . m)))) . g is complex set
(f +* (m,T)) . g is complex set
f . g is complex set
((f +* (m,T)) . g) - (f . g) is complex set
- (f . g) is complex set
((f +* (m,T)) . g) + (- (f . g)) is complex set
T - (f . g) is complex set
T + (- (f . g)) is complex set
(f . g) - (f . g) is complex set
(f . g) + (- (f . g)) is complex set
(r |-> 0) . g is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative ordinal natural complex complex-valued ext-real-valued real-valued natural-valued real finite finite-yielding V49() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V132() Function-yielding V156() V157() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued V267() V268() set
m is ext-real complex real set
<*m*> is Relation-like NAT -defined Function-like one-to-one constant non empty trivial complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing finite 1 -element FinSequence-like FinSubsequence-like V268() set
[1,m] is set
{1,m} is non empty finite complex-membered ext-real-membered real-membered set
{{1,m},{1}} is non empty finite V49() set
{[1,m]} is Relation-like Function-like constant non empty trivial finite 1 -element V268() set
|.<*m*>.| is ext-real non negative complex real Element of REAL
sqr <*m*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty trivial complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing finite 1 -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
<*m*> (#) <*m*> is Relation-like NAT -defined Function-like one-to-one constant non empty trivial complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing finite 1 -element FinSequence-like FinSubsequence-like V268() set
K857() .: (<*m*>,<*m*>) is Relation-like Function-like set
<:<*m*>,<*m*>:> is Relation-like Function-like set
<:<*m*>,<*m*>:> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
<*m*> (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum (sqr <*m*>) is ext-real complex real Element of REAL
sqrt (Sum (sqr <*m*>)) is ext-real complex real Element of REAL
abs m is ext-real complex real Element of REAL
m ^2 is ext-real non negative complex real set
m * m is ext-real non negative complex real set
<*(m ^2)*> is Relation-like NAT -defined Function-like one-to-one constant non empty trivial complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing finite 1 -element FinSequence-like FinSubsequence-like V268() set
[1,(m ^2)] is set
{1,(m ^2)} is non empty finite complex-membered ext-real-membered real-membered set
{{1,(m ^2)},{1}} is non empty finite V49() set
{[1,(m ^2)]} is Relation-like Function-like constant non empty trivial finite 1 -element V268() set
m is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
m is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
|.m.| is ext-real non negative complex real Element of REAL
sqr m is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() FinSequence of REAL
m (#) m is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
K857() .: (m,m) is Relation-like Function-like set
<:m,m:> is Relation-like Function-like set
<:m,m:> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
m (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum (sqr m) is ext-real complex real Element of REAL
sqrt (Sum (sqr m)) is ext-real complex real Element of REAL
dom m is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom (sqr m) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
r is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
(sqr m) . r is ext-real complex real set
m . r is ext-real complex real set
(m . r) ^2 is ext-real non negative complex real set
(m . r) * (m . r) is ext-real non negative complex real set
m is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum m is ext-real complex real set
abs (Sum m) is ext-real complex real Element of REAL
abs m is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() FinSequence of REAL
absreal is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
m (#) absreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum (abs m) is ext-real complex real Element of REAL
Sum (<*> REAL) is ext-real complex real set
abs (Sum (<*> REAL)) is ext-real complex real Element of REAL
abs (<*> REAL) is Relation-like non-empty empty-yielding NAT -defined REAL -valued Function-like one-to-one constant functional empty ext-real non positive non negative ordinal natural complex complex-valued ext-real-valued real-valued natural-valued real finite finite-yielding V49() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V132() Function-yielding V156() V157() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued V267() V268() FinSequence of REAL
(<*> REAL) (#) absreal is Relation-like non-empty empty-yielding NAT -defined REAL -valued RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative ordinal natural complex complex-valued ext-real-valued real-valued natural-valued real finite finite-yielding V49() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V132() Function-yielding V156() V157() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued V267() V268() set
Sum (abs (<*> REAL)) is ext-real complex real Element of REAL
r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() FinSequence of REAL
T is ext-real complex real Element of REAL
<*T*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty trivial complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing finite 1 -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
[1,T] is set
{1,T} is non empty finite complex-membered ext-real-membered real-membered set
{{1,T},{1}} is non empty finite V49() set
{[1,T]} is Relation-like Function-like constant non empty trivial finite 1 -element V268() set
r ^ <*T*> is Relation-like NAT -defined REAL -valued Function-like non empty complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() FinSequence of REAL
f is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum f is ext-real complex real set
abs (Sum f) is ext-real complex real Element of REAL
abs f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() FinSequence of REAL
f (#) absreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum (abs f) is ext-real complex real Element of REAL
abs T is ext-real complex real Element of REAL
(abs (Sum f)) + (abs T) is ext-real complex real set
(Sum (abs f)) + (abs T) is ext-real complex real set
g is Relation-like NAT -defined REAL -valued Function-like non empty complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() FinSequence of REAL
Sum g is ext-real complex real set
abs (Sum g) is ext-real complex real Element of REAL
abs g is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() FinSequence of REAL
g (#) absreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum (abs g) is ext-real complex real Element of REAL
abs <*T*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty trivial complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing finite 1 -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
<*T*> (#) absreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
<*(abs T)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty trivial complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing finite 1 -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
[1,(abs T)] is set
{1,(abs T)} is non empty finite complex-membered ext-real-membered real-membered set
{{1,(abs T)},{1}} is non empty finite V49() set
{[1,(abs T)]} is Relation-like Function-like constant non empty trivial finite 1 -element V268() set
Sum <*(abs T)*> is ext-real complex real Element of REAL
abs r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() FinSequence of REAL
r (#) absreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
(abs r) ^ (abs <*T*>) is Relation-like NAT -defined REAL -valued Function-like non empty complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() FinSequence of REAL
Sum (abs r) is ext-real complex real Element of REAL
(Sum (abs r)) + (abs T) is ext-real complex real set
Sum g is ext-real complex real Element of REAL
Sum r is ext-real complex real Element of REAL
(Sum r) + T is ext-real complex real set
abs (Sum g) is ext-real complex real Element of REAL
abs (Sum r) is ext-real complex real Element of REAL
(abs (Sum r)) + (abs T) is ext-real complex real set
r is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum r is ext-real complex real set
abs (Sum r) is ext-real complex real Element of REAL
abs r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() FinSequence of REAL
r (#) absreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum (abs r) is ext-real complex real Element of REAL
m is ext-real non negative ordinal natural complex real finite cardinal set
m -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V268() Element of REAL * : len b₁ = m } is set
r is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
len r is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
m is non empty 1-sorted
the carrier of m is non empty set
r is non empty trivial finite 1 -element 1-sorted
the carrier of r is non empty trivial finite 1 -element set
[: the carrier of m, the carrier of r:] is Relation-like set
bool [: the carrier of m, the carrier of r:] is set
T is Element of the carrier of r
m --> T is Relation-like the carrier of m -defined the carrier of r -valued Function-like non empty total V18( the carrier of m, the carrier of r) Element of bool [: the carrier of m, the carrier of r:]
the carrier of m --> T is Relation-like the carrier of m -defined the carrier of r -valued Function-like constant non empty total V18( the carrier of m, the carrier of r) Element of bool [: the carrier of m, the carrier of r:]
{T} is non empty trivial finite 1 -element set
[: the carrier of m,{T}:] is Relation-like set
f is Relation-like the carrier of m -defined the carrier of r -valued Function-like non empty total V18( the carrier of m, the carrier of r) Element of bool [: the carrier of m, the carrier of r:]
g is Element of the carrier of m
f . g is set
(m --> T) . g is set
f . g is Element of the carrier of r
(m --> T) . g is Element of the carrier of r
m is non empty ext-real positive non negative ordinal natural complex real finite cardinal set
Seg m is non empty finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
T is non empty TopSpace-like real-membered TopStruct
(Seg m) --> T is Relation-like Seg m -defined Seg m -defined {T} -valued Function-like constant non empty total total V18( Seg m,{T}) finite FinSequence-like FinSubsequence-like V268() V291() V306() V309() Element of bool [:(Seg m),{T}:]
{T} is non empty trivial finite 1 -element set
[:(Seg m),{T}:] is Relation-like finite set
bool [:(Seg m),{T}:] is finite V49() set
r is ext-real non negative ordinal natural complex real V43() V44() finite cardinal Element of Seg m
proj (((Seg m) --> T),r) is Relation-like the carrier of (product ((Seg m) --> T)) -defined the carrier of (((Seg m) --> T) . r) -valued Function-like non empty total V18( the carrier of (product ((Seg m) --> T)), the carrier of (((Seg m) --> T) . r)) Element of bool [: the carrier of (product ((Seg m) --> T)), the carrier of (((Seg m) --> T) . r):]
product ((Seg m) --> T) is non empty V138() strict TopSpace-like TopStruct
the carrier of (product ((Seg m) --> T)) is non empty set
((Seg m) --> T) . r is non empty TopStruct
the carrier of (((Seg m) --> T) . r) is non empty set
[: the carrier of (product ((Seg m) --> T)), the carrier of (((Seg m) --> T) . r):] is Relation-like set
bool [: the carrier of (product ((Seg m) --> T)), the carrier of (((Seg m) --> T) . r):] is set
m is ext-real non negative ordinal natural complex real finite cardinal set
REAL m is functional non empty FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V268() Element of REAL * : len b₁ = m } is set
r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of REAL m
T is ext-real complex real set
r (/) T is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
1 / T is ext-real complex real set
T " is ext-real complex real set
1 * (T ") is ext-real complex real set
(1 / T) (#) r is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K260((1 / T)) is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
id REAL is Relation-like REAL -defined REAL -valued Function-like one-to-one non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued increasing non-decreasing Element of bool [:REAL,REAL:]
K857() [;] ((1 / T),(id REAL)) is Relation-like Function-like set
dom (id REAL) is non empty set
(dom (id REAL)) --> (1 / T) is Relation-like dom (id REAL) -defined Function-like constant non empty total complex-valued ext-real-valued real-valued set
{(1 / T)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(dom (id REAL)),{(1 / T)}:] is Relation-like complex-valued ext-real-valued real-valued set
<:((dom (id REAL)) --> (1 / T)),(id REAL):> is Relation-like Function-like set
<:((dom (id REAL)) --> (1 / T)),(id REAL):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
r (#) K260((1 / T)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
len r is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
len (r (/) T) is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
Seg (len (r (/) T)) is finite len (r (/) T) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= len (r (/) T) ) } is set
dom (r (/) T) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom r is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
Seg m is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict L16()
the carrier of (TOP-REAL m) is non empty set
r is ext-real complex real set
T is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
Ball (f,r) is open V269( TOP-REAL m) Element of bool the carrier of (TOP-REAL m)
bool the carrier of (TOP-REAL m) is set
- T is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- T is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) T is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K260((- 1)) is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K857() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined Function-like constant non empty total complex-valued ext-real-valued real-valued V315() V316() set
{(- 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(dom (id REAL)),{(- 1)}:] is Relation-like complex-valued ext-real-valued real-valued set
<:((dom (id REAL)) --> (- 1)),(id REAL):> is Relation-like Function-like set
<:((dom (id REAL)) --> (- 1)),(id REAL):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
T (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
K853() is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
T (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
- f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(- 1) (#) f is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
f (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
f (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Ball ((- f),r) is open V269( TOP-REAL m) Element of bool the carrier of (TOP-REAL m)
g is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
p is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
Ball (p,r) is open V269( TOP-REAL m) Element of bool the carrier of (TOP-REAL m)
g - p is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- p is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- p is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(- 1) (#) p is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
p (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
p (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
g + (- p) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
g + (- p) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K855() .: (g,(- p)) is Relation-like Function-like set
g - p is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
- p is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
g + (- p) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K855() .: (g,(- p)) is Relation-like Function-like set
K856() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K342(REAL,K855(),(id REAL),K853()) is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K856() .: (g,p) is Relation-like Function-like set
<:g,p:> is Relation-like Function-like set
<:g,p:> (#) K856() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
|.(g - p).| is ext-real non negative complex real Element of REAL
sqr (g - p) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(g - p) (#) (g - p) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K857() .: ((g - p),(g - p)) is Relation-like Function-like set
<:(g - p),(g - p):> is Relation-like Function-like set
<:(g - p),(g - p):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(g - p) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum (sqr (g - p)) is ext-real complex real Element of REAL
sqrt (Sum (sqr (g - p))) is ext-real complex real Element of REAL
- g is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- g is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(- 1) (#) g is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
g (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
g (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
(- g) - (- p) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- (- p) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- (- p) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(- 1) (#) (- p) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
(- p) (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
(- p) (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
(- g) + (- (- p)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(- g) + (- (- p)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() .: ((- g),(- (- p))) is Relation-like Function-like set
(- g) - (- p) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
- (- p) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
(- g) + (- (- p)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K855() .: ((- g),(- (- p))) is Relation-like Function-like set
K856() .: ((- g),(- p)) is Relation-like Function-like set
<:(- g),(- p):> is Relation-like Function-like set
<:(- g),(- p):> (#) K856() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(- g) + (- (- p)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- (g - p) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- (g - p) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(- 1) (#) (g - p) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
(g - p) (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
(g - p) (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
|.((- g) - (- p)).| is ext-real non negative complex real Element of REAL
sqr ((- g) - (- p)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
((- g) - (- p)) (#) ((- g) - (- p)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K857() .: (((- g) - (- p)),((- g) - (- p))) is Relation-like Function-like set
<:((- g) - (- p)),((- g) - (- p)):> is Relation-like Function-like set
<:((- g) - (- p)),((- g) - (- p)):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
((- g) - (- p)) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum (sqr ((- g) - (- p))) is ext-real complex real Element of REAL
sqrt (Sum (sqr ((- g) - (- p)))) is ext-real complex real Element of REAL
Ball ((- p),r) is open V269( TOP-REAL m) Element of bool the carrier of (TOP-REAL m)
- (- T) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- (- T) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(- 1) (#) (- T) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
(- T) (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
(- T) (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
- (- f) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- (- f) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(- 1) (#) (- f) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
(- f) (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
(- f) (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict L16()
r is set
the carrier of (TOP-REAL m) is non empty set
T is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
m is 1-sorted
the carrier of m is set
r is set
the Relation-like Function-like complex-valued ext-real-valued real-valued set is Relation-like Function-like complex-valued ext-real-valued real-valued set
{ the Relation-like Function-like complex-valued ext-real-valued real-valued set } is functional non empty trivial finite 1 -element complex-functions-membered ext-real-functions-membered real-functions-membered set
1-sorted(# { the Relation-like Function-like complex-valued ext-real-valued real-valued set } #) is strict 1-sorted
m is strict 1-sorted
the carrier of m is set
r is set
m is () 1-sorted
the carrier of m is set
m is () () 1-sorted
the carrier of m is functional complex-functions-membered set
TOP-REAL 1 is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL 1) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
the topology of (TOP-REAL 1) is non empty Element of bool (bool the carrier of (TOP-REAL 1))
bool the carrier of (TOP-REAL 1) is set
bool (bool the carrier of (TOP-REAL 1)) is set
TopStruct(# the carrier of (TOP-REAL 1), the topology of (TOP-REAL 1) #) is non empty strict TopSpace-like TopStruct
m is TopSpace-like () TopStruct
r is TopSpace-like SubSpace of m
T is set
the carrier of r is set
the carrier of m is functional complex-functions-membered set
m is TopSpace-like () () TopStruct
r is TopSpace-like () SubSpace of m
T is set
the carrier of r is functional complex-functions-membered set
the carrier of m is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
m is functional complex-functions-membered set
{ (- b₁) where b₁ is Relation-like Function-like complex-valued Element of m : b₁ in m } is set
T is set
f is Relation-like Function-like complex-valued Element of m
- f is Relation-like Function-like complex-valued set
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) f is Relation-like Function-like complex-valued set
T is functional complex-functions-membered set
f is Relation-like Function-like complex-valued set
- f is Relation-like Function-like complex-valued set
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) f is Relation-like Function-like complex-valued set
g is Relation-like Function-like complex-valued Element of m
- g is Relation-like Function-like complex-valued set
(- 1) (#) g is Relation-like Function-like complex-valued set
r is functional complex-functions-membered set
T is functional complex-functions-membered set
f is set
g is Relation-like Function-like complex-valued Element of r
- g is Relation-like Function-like complex-valued set
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) g is Relation-like Function-like complex-valued set
- (- g) is Relation-like Function-like complex-valued set
(- 1) (#) (- g) is Relation-like Function-like complex-valued set
f is set
g is Relation-like Function-like complex-valued Element of T
- g is Relation-like Function-like complex-valued set
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) g is Relation-like Function-like complex-valued set
- (- g) is Relation-like Function-like complex-valued set
(- 1) (#) (- g) is Relation-like Function-like complex-valued set
r is functional complex-functions-membered set
T is functional complex-functions-membered set
f is Relation-like Function-like complex-valued set
- f is Relation-like Function-like complex-valued set
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) 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
m is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative ordinal natural complex complex-valued ext-real-valued real-valued natural-valued real finite finite-yielding V49() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V132() Function-yielding V156() V157() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued V267() V268() set
(m) is functional complex-functions-membered set
r is set
T is Relation-like Function-like complex-valued Element of (m)
- T is Relation-like Function-like complex-valued set
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) T is Relation-like Function-like complex-valued set
- (- T) is Relation-like Function-like complex-valued set
(- 1) (#) (- T) is Relation-like Function-like complex-valued set
m is functional non empty complex-functions-membered set
(m) is functional complex-functions-membered set
r is set
T is Relation-like Function-like complex-valued Element of m
- T is Relation-like Function-like complex-valued set
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) T is Relation-like Function-like complex-valued set
m is functional complex-functions-membered set
(m) is functional complex-functions-membered set
r is Relation-like Function-like complex-valued set
- r is Relation-like Function-like complex-valued set
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) r is Relation-like Function-like complex-valued set
- (- r) is Relation-like Function-like complex-valued set
(- 1) (#) (- r) is Relation-like Function-like complex-valued set
((m)) is functional complex-functions-membered set
m is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
(m) is functional complex-functions-membered set
r is set
T is Relation-like Function-like complex-valued Element of (m)
- T is Relation-like Function-like complex-valued set
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) T is Relation-like Function-like complex-valued set
- (- T) is Relation-like Function-like complex-valued set
(- 1) (#) (- T) is Relation-like Function-like complex-valued set
((m)) is functional complex-functions-membered set
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
bool the carrier of (TOP-REAL m) is set
T is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL m)
- T is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL m)
K734((TOP-REAL m),T,K504(1)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL m)
{ (K504(1) * b₁) where b₁ is Element of the carrier of (TOP-REAL m) : b₁ in T } is set
(T) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
f is Relation-like Function-like complex-valued set
- f is Relation-like Function-like complex-valued set
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) f is Relation-like Function-like complex-valued set
g is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(- 1) * g is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
the Mult of (TOP-REAL m) is Relation-like [:REAL, the carrier of (TOP-REAL m):] -defined the carrier of (TOP-REAL m) -valued Function-like total V18([:REAL, the carrier of (TOP-REAL m):], the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:[:REAL, the carrier of (TOP-REAL m):], the carrier of (TOP-REAL m):]
[:REAL, the carrier of (TOP-REAL m):] is Relation-like non trivial non finite set
[:[:REAL, the carrier of (TOP-REAL m):], the carrier of (TOP-REAL m):] is Relation-like non trivial non finite set
bool [:[:REAL, the carrier of (TOP-REAL m):], the carrier of (TOP-REAL m):] is non trivial non finite set
the Mult of (TOP-REAL m) . ((- 1),g) is set
[(- 1),g] is set
{(- 1),g} is non empty finite set
{(- 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
{{(- 1),g},{(- 1)}} is non empty finite V49() set
the Mult of (TOP-REAL m) . [(- 1),g] is Relation-like Function-like complex-valued ext-real-valued real-valued set
(- 1) * g is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K260((- 1)) is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K857() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined Function-like constant non empty total complex-valued ext-real-valued real-valued V315() V316() set
[:(dom (id REAL)),{(- 1)}:] is Relation-like complex-valued ext-real-valued real-valued set
<:((dom (id REAL)) --> (- 1)),(id REAL):> is Relation-like Function-like set
<:((dom (id REAL)) --> (- 1)),(id REAL):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
g (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
p is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- p is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- p is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(- 1) (#) p is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
p (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
K853() is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
p (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
- g is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- g is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(- 1) (#) g is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
g (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
- (- g) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- (- g) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(- 1) (#) (- g) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
(- g) (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
(- g) (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
g is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- g is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- g is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(- 1) (#) g is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
g (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
g (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
(- 1) * g is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
the Mult of (TOP-REAL m) . ((- 1),g) is set
[(- 1),g] is set
{(- 1),g} is non empty finite set
{{(- 1),g},{(- 1)}} is non empty finite V49() set
the Mult of (TOP-REAL m) . [(- 1),g] is Relation-like Function-like complex-valued ext-real-valued real-valued set
(- 1) * g is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
bool the carrier of (TOP-REAL m) is set
r is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL m)
(r) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
- r is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL m)
K734((TOP-REAL m),r,K504(1)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL m)
{ (K504(1) * b₁) where b₁ is Element of the carrier of (TOP-REAL m) : b₁ in r } is set
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
bool the carrier of (TOP-REAL m) is set
r is functional complex-functions-membered ext-real-functions-membered real-functions-membered open Element of bool the carrier of (TOP-REAL m)
(m,r) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL m)
T is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
TOP-REAL T is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL T) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
bool the carrier of (TOP-REAL T) is set
f is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL T)
g is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite T -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL T)
- g is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite T -element FinSequence-like FinSubsequence-like V268() set
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) g is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite T -element FinSequence-like FinSubsequence-like V268() set
K260((- 1)) is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K857() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined Function-like constant non empty total complex-valued ext-real-valued real-valued V315() V316() set
{(- 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(dom (id REAL)),{(- 1)}:] is Relation-like complex-valued ext-real-valued real-valued set
<:((dom (id REAL)) --> (- 1)),(id REAL):> is Relation-like Function-like set
<:((dom (id REAL)) --> (- 1)),(id REAL):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
g (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
K853() is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
g (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
(T,f) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL T)
- g is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite T -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL T)
- g is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite T -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
p is non empty ext-real positive non negative complex real set
Ball ((- g),p) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered open V215( TOP-REAL T) V269( TOP-REAL T) Element of bool the carrier of (TOP-REAL T)
Ball (g,p) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered open V215( TOP-REAL T) V269( TOP-REAL T) Element of bool the carrier of (TOP-REAL T)
r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite T -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL T)
- r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite T -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL T)
- r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite T -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(- 1) (#) r is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite T -element FinSequence-like FinSubsequence-like V268() set
r (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
r (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
g is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL m)
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
T is set
r . T is ext-real complex real set
m is non empty TopSpace-like TopStruct
r is non empty TopSpace-like TopStruct
[:m,r:] is non empty strict TopSpace-like TopStruct
the carrier of [:m,r:] is non empty set
T is non empty TopSpace-like TopStruct
the carrier of T is non empty set
[: the carrier of [:m,r:], the carrier of T:] is Relation-like set
bool [: the carrier of [:m,r:], the carrier of T:] is set
f is Relation-like the carrier of [:m,r:] -defined the carrier of T -valued Function-like non empty total V18( the carrier of [:m,r:], the carrier of T) Element of bool [: the carrier of [:m,r:], the carrier of T:]
g is Element of the carrier of [:m,r:]
f . g is set
the carrier of m is non empty set
the carrier of r is non empty set
p is Element of the carrier of m
r is Element of the carrier of r
[p,r] is Element of the carrier of [:m,r:]
{p,r} is non empty finite set
{p} is non empty trivial finite 1 -element set
{{p,r},{p}} is non empty finite V49() set
f . [p,r] is Element of the carrier of T
m is non empty TopSpace-like TopStruct
r is non empty TopSpace-like TopStruct
[:m,r:] is non empty strict TopSpace-like TopStruct
the carrier of [:m,r:] is non empty set
T is non empty TopSpace-like TopStruct
the carrier of T is non empty set
[: the carrier of [:m,r:], the carrier of T:] is Relation-like set
bool [: the carrier of [:m,r:], the carrier of T:] is set
the carrier of m is non empty set
the carrier of r is non empty set
f is Relation-like the carrier of [:m,r:] -defined the carrier of T -valued Function-like non empty total V18( the carrier of [:m,r:], the carrier of T) Element of bool [: the carrier of [:m,r:], the carrier of T:]
g is Element of the carrier of m
p is Element of the carrier of r
f . (g,p) is set
[g,p] is set
{g,p} is non empty finite set
{g} is non empty trivial finite 1 -element set
{{g,p},{g}} is non empty finite V49() set
f . [g,p] is set
[g,p] is Element of the carrier of [:m,r:]
(m,r,T,f,[g,p]) is Element of the carrier of T
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
T is ext-real complex real set
T + r is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
len r is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
len (T + r) is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
Seg (len (T + r)) is finite len (T + r) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= len (T + r) ) } is set
dom (T + r) is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom r is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
Seg m is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
REAL m is functional non empty FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V268() Element of REAL * : len b₁ = m } is set
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
T is ext-real complex real set
r - T is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
- T is ext-real complex real set
(- T) + r is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
(m,r,(- T)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
T is ext-real complex real set
T (#) r is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K260(T) is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K857() [;] (T,(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> T is Relation-like dom (id REAL) -defined Function-like constant non empty total complex-valued ext-real-valued real-valued set
{T} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(dom (id REAL)),{T}:] is Relation-like complex-valued ext-real-valued real-valued set
<:((dom (id REAL)) --> T),(id REAL):> is Relation-like Function-like set
<:((dom (id REAL)) --> T),(id REAL):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
r (#) K260(T) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
len r is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
len (T (#) r) is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
Seg (len (T (#) r)) is finite len (T (#) r) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= len (T (#) r) ) } is set
dom (T (#) r) is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom r is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
Seg m is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
REAL m is functional non empty FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V268() Element of REAL * : len b₁ = m } is set
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
T is ext-real complex real set
r (/) T is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
1 / T is ext-real complex real set
T " is ext-real complex real set
1 * (T ") is ext-real complex real set
(1 / T) (#) r is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K260((1 / T)) is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K857() [;] ((1 / T),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (1 / T) is Relation-like dom (id REAL) -defined Function-like constant non empty total complex-valued ext-real-valued real-valued set
{(1 / T)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(dom (id REAL)),{(1 / T)}:] is Relation-like complex-valued ext-real-valued real-valued set
<:((dom (id REAL)) --> (1 / T)),(id REAL):> is Relation-like Function-like set
<:((dom (id REAL)) --> (1 / T)),(id REAL):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
r (#) K260((1 / T)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
REAL m is functional non empty FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V268() Element of REAL * : len b₁ = m } is set
f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of REAL m
(m,f,T) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of REAL m
(1 / T) (#) f is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
f (#) K260((1 / T)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
T is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
r (#) T is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K857() .: (r,T) is Relation-like Function-like set
<:r,T:> is Relation-like Function-like set
<:r,T:> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
len r is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
len T is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
len (r (#) T) is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
Seg (len (r (#) T)) is finite len (r (#) T) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= len (r (#) T) ) } is set
dom (r (#) T) is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom r is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom T is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
(dom r) /\ (dom T) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
Seg m is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
(Seg m) /\ (dom T) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
(Seg m) /\ (Seg m) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
REAL m is functional non empty FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V268() Element of REAL * : len b₁ = m } is set
f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
g is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
f (#) g is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K857() .: (f,g) is Relation-like Function-like set
<:f,g:> is Relation-like Function-like set
<:f,g:> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
g (#) f is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K857() .: (g,f) is Relation-like Function-like set
<:g,f:> is Relation-like Function-like set
<:g,f:> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
r ^2 is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
r (#) r is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K857() .: (r,r) is Relation-like Function-like set
<:r,r:> is Relation-like Function-like set
<:r,r:> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
r (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
sqr r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(m,r,r) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
T is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
r /" T is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
T " is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
r (#) (T ") is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K857() .: (r,(T ")) is Relation-like Function-like set
<:r,(T "):> is Relation-like Function-like set
<:r,(T "):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
len r is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
len T is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
len (r /" T) is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
Seg (len (r /" T)) is finite len (r /" T) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= len (r /" T) ) } is set
dom (r /" T) is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom r is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom T is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
(dom r) /\ (dom T) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
Seg m is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
(Seg m) /\ (dom T) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
(Seg m) /\ (Seg m) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
REAL m is functional non empty FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V268() Element of REAL * : len b₁ = m } is set
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
T is set
f is ext-real complex real set
r +* (T,f) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
len r is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
len (r +* (T,f)) is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
Seg (len (r +* (T,f))) is finite len (r +* (T,f)) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= len (r +* (T,f)) ) } is set
dom (r +* (T,f)) is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom r is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
Seg m is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
REAL m is functional non empty FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V268() Element of REAL * : len b₁ = m } is set
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
r is ext-real complex real set
m * r is ext-real complex real set
T is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
Ball (f,r) is functional complex-functions-membered ext-real-functions-membered real-functions-membered open V269( TOP-REAL m) Element of bool the carrier of (TOP-REAL m)
bool the carrier of (TOP-REAL m) is set
T - f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) f is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K260((- 1)) is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K857() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined Function-like constant non empty total complex-valued ext-real-valued real-valued V315() V316() set
{(- 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(dom (id REAL)),{(- 1)}:] is Relation-like complex-valued ext-real-valued real-valued set
<:((dom (id REAL)) --> (- 1)),(id REAL):> is Relation-like Function-like set
<:((dom (id REAL)) --> (- 1)),(id REAL):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
f (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
K853() is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
f (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
T + (- f) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
T + (- f) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K855() .: (T,(- f)) is Relation-like Function-like set
T - f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
- f is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
T + (- f) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K855() .: (T,(- f)) is Relation-like Function-like set
K856() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K342(REAL,K855(),(id REAL),K853()) is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K856() .: (T,f) is Relation-like Function-like set
<:T,f:> is Relation-like Function-like set
<:T,f:> (#) K856() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
Sum (T - f) is ext-real complex real set
abs (Sum (T - f)) is ext-real complex real Element of REAL
m |-> r is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
Seg m is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
(Seg m) --> r is Relation-like Seg m -defined Seg m -defined {r} -valued Function-like constant total total V18( Seg m,{r}) complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() Element of bool [:(Seg m),{r}:]
{r} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(Seg m),{r}:] is Relation-like complex-valued ext-real-valued real-valued finite set
bool [:(Seg m),{r}:] is finite V49() set
Sum (m |-> r) is ext-real complex real set
abs (T - f) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
absreal is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
(T - f) (#) absreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
m -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V268() Element of REAL * : len b₁ = m } is set
r is ext-real non negative ordinal natural complex real finite cardinal set
(m |-> r) . r is ext-real complex real set
(abs (T - f)) . r is ext-real complex real set
(m,(T - f),r) is ext-real complex real Element of REAL
abs (m,(T - f),r) is ext-real complex real Element of REAL
r is ext-real non negative ordinal natural complex real finite cardinal set
(abs (T - f)) . r is ext-real complex real set
(m |-> r) . r is ext-real complex real set
(m |-> r) . 1 is ext-real complex real set
(abs (T - f)) . 1 is ext-real complex real set
Sum (abs (T - f)) is ext-real complex real Element of REAL
r is ext-real non negative ordinal natural complex real finite cardinal set
(m |-> r) . r is ext-real complex real set
(abs (T - f)) . r is ext-real complex real set
m is ext-real non negative ordinal natural complex real finite cardinal set
Euclid m is non empty V138() V139() strict Reflexive discerning V193() triangle MetrStruct
REAL m is functional non empty FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V268() Element of REAL * : len b₁ = m } is set
Pitag_dist m is Relation-like [:(REAL m),(REAL m):] -defined REAL -valued Function-like total V18([:(REAL m),(REAL m):], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:(REAL m),(REAL m):],REAL:]
[:(REAL m),(REAL m):] is Relation-like set
[:[:(REAL m),(REAL m):],REAL:] is Relation-like complex-valued ext-real-valued real-valued set
bool [:[:(REAL m),(REAL m):],REAL:] is set
MetrStruct(# (REAL m),(Pitag_dist m) #) is strict MetrStruct
r is set
the carrier of (Euclid m) is non empty set
T is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (Euclid m)
m is non empty right_complementable add-associative right_zeroed addLoopStr
the carrier of m is non empty set
r is Element of the carrier of m
T is Element of the carrier of m
r + T is Element of the carrier of m
(r + T) - T is Element of the carrier of m
- T is Element of the carrier of m
(r + T) + (- T) is Element of the carrier of m
T - T is Element of the carrier of m
T + (- T) is Element of the carrier of m
r + (T - T) is Element of the carrier of m
0. m is zero Element of the carrier of m
the ZeroF of m is Element of the carrier of m
r + (0. m) is Element of the carrier of m
m is non empty left_complementable right_complementable Abelian add-associative right_zeroed zeroed addLoopStr
the carrier of m is non empty set
r is Element of the carrier of m
T is Element of the carrier of m
r - T is Element of the carrier of m
- T is Element of the carrier of m
r + (- T) is Element of the carrier of m
(r - T) + T is Element of the carrier of m
T - T is Element of the carrier of m
T + (- T) is Element of the carrier of m
r - (T - T) is Element of the carrier of m
- (T - T) is Element of the carrier of m
r + (- (T - T)) is Element of the carrier of m
0. m is zero Element of the carrier of m
the ZeroF of m is Element of the carrier of m
r - (0. m) is Element of the carrier of m
- (0. m) is Element of the carrier of m
r + (- (0. m)) is Element of the carrier of m
m is set
r is complex set
T is functional complex-functions-membered set
[:m,T:] is Relation-like set
bool [:m,T:] is set
f is Relation-like m -defined T -valued Function-like Function-yielding V156() complex-functions-valued Element of bool [:m,T:]
f [+] r is Relation-like m -defined C_PFuncs (DOMS T) -valued Function-like Function-yielding V156() complex-functions-valued Element of bool [:m,(C_PFuncs (DOMS T)):]
DOMS T is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of T : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of T : verum } is set
C_PFuncs (DOMS T) is functional complex-functions-membered set
[:m,(C_PFuncs (DOMS T)):] is Relation-like set
bool [:m,(C_PFuncs (DOMS T)):] is set
dom f is Element of bool m
bool m is set
(dom f) --> r is Relation-like dom f -defined {r} -valued Function-like constant total V18( dom f,{r}) complex-valued Element of bool [:(dom f),{r}:]
{r} is non empty trivial finite 1 -element complex-membered set
[:(dom f),{r}:] is Relation-like complex-valued set
bool [:(dom f),{r}:] is set
f <+> ((dom f) --> r) is Relation-like m /\ (dom ((dom f) --> r)) -defined C_PFuncs (DOMS T) -valued Function-like Function-yielding V156() complex-functions-valued Element of bool [:(m /\ (dom ((dom f) --> r))),(C_PFuncs (DOMS T)):]
dom ((dom f) --> r) is set
m /\ (dom ((dom f) --> r)) is set
[:(m /\ (dom ((dom f) --> r))),(C_PFuncs (DOMS T)):] is Relation-like set
bool [:(m /\ (dom ((dom f) --> r))),(C_PFuncs (DOMS T)):] is set
dom (f [+] r) is Element of bool m
dom (f <+> ((dom f) --> r)) is Element of bool (m /\ (dom ((dom f) --> r)))
bool (m /\ (dom ((dom f) --> r))) is set
dom ((dom f) --> r) is Element of bool (dom f)
bool (dom f) is set
(dom f) /\ (dom ((dom f) --> r)) is Element of bool (dom f)
p is set
(f [+] r) . p is Relation-like Function-like complex-valued set
f . p is Relation-like Function-like complex-valued set
r + (f . p) is Relation-like Function-like complex-valued set
((dom f) --> r) . p is complex set
(((dom f) --> r) . p) + (f . p) is Relation-like Function-like complex-valued set
(f <+> ((dom f) --> r)) . p is Relation-like Function-like complex-valued set
m is set
r is complex set
T is functional complex-functions-membered set
[:m,T:] is Relation-like set
bool [:m,T:] is set
f is Relation-like m -defined T -valued Function-like Function-yielding V156() complex-functions-valued Element of bool [:m,T:]
f [-] r is Relation-like m -defined C_PFuncs (DOMS T) -valued Function-like Function-yielding V156() complex-functions-valued Element of bool [:m,(C_PFuncs (DOMS T)):]
DOMS T is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of T : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of T : verum } is set
C_PFuncs (DOMS T) is functional complex-functions-membered set
[:m,(C_PFuncs (DOMS T)):] is Relation-like set
bool [:m,(C_PFuncs (DOMS T)):] is set
- r is complex set
f [+] (- r) is Relation-like Function-like set
dom f is Element of bool m
bool m is set
(dom f) --> r is Relation-like dom f -defined {r} -valued Function-like constant total V18( dom f,{r}) complex-valued Element of bool [:(dom f),{r}:]
{r} is non empty trivial finite 1 -element complex-membered set
[:(dom f),{r}:] is Relation-like complex-valued set
bool [:(dom f),{r}:] is set
f <-> ((dom f) --> r) is Relation-like m /\ (dom ((dom f) --> r)) -defined C_PFuncs (DOMS T) -valued Function-like Function-yielding V156() complex-functions-valued Element of bool [:(m /\ (dom ((dom f) --> r))),(C_PFuncs (DOMS T)):]
dom ((dom f) --> r) is set
m /\ (dom ((dom f) --> r)) is set
[:(m /\ (dom ((dom f) --> r))),(C_PFuncs (DOMS T)):] is Relation-like set
bool [:(m /\ (dom ((dom f) --> r))),(C_PFuncs (DOMS T)):] is set
- ((dom f) --> r) is Relation-like dom f -defined Function-like total complex-valued set
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) ((dom f) --> r) is Relation-like dom f -defined Function-like total complex-valued set
f <+> (- ((dom f) --> r)) is Relation-like Function-like set
dom (f [-] r) is Element of bool m
dom (f <-> ((dom f) --> r)) is Element of bool (m /\ (dom ((dom f) --> r)))
bool (m /\ (dom ((dom f) --> r))) is set
dom ((dom f) --> r) is Element of bool (dom f)
bool (dom f) is set
(dom f) /\ (dom ((dom f) --> r)) is Element of bool (dom f)
p is set
(f [-] r) . p is Relation-like Function-like complex-valued set
f . p is Relation-like Function-like complex-valued set
(f . p) - r is Relation-like Function-like complex-valued set
(- r) + (f . p) is Relation-like Function-like complex-valued set
((dom f) --> r) . p is complex set
(f . p) - (((dom f) --> r) . p) is Relation-like Function-like complex-valued set
- (((dom f) --> r) . p) is complex set
(- (((dom f) --> r) . p)) + (f . p) is Relation-like Function-like complex-valued set
(f <-> ((dom f) --> r)) . p is Relation-like Function-like complex-valued set
m is set
r is complex set
T is functional complex-functions-membered set
[:m,T:] is Relation-like set
bool [:m,T:] is set
f is Relation-like m -defined T -valued Function-like Function-yielding V156() complex-functions-valued Element of bool [:m,T:]
f [#] r is Relation-like m -defined C_PFuncs (DOMS T) -valued Function-like Function-yielding V156() complex-functions-valued Element of bool [:m,(C_PFuncs (DOMS T)):]
DOMS T is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of T : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of T : verum } is set
C_PFuncs (DOMS T) is functional complex-functions-membered set
[:m,(C_PFuncs (DOMS T)):] is Relation-like set
bool [:m,(C_PFuncs (DOMS T)):] is set
dom f is Element of bool m
bool m is set
(dom f) --> r is Relation-like dom f -defined {r} -valued Function-like constant total V18( dom f,{r}) complex-valued Element of bool [:(dom f),{r}:]
{r} is non empty trivial finite 1 -element complex-membered set
[:(dom f),{r}:] is Relation-like complex-valued set
bool [:(dom f),{r}:] is set
f <#> ((dom f) --> r) is Relation-like m /\ (dom ((dom f) --> r)) -defined C_PFuncs (DOMS T) -valued Function-like Function-yielding V156() complex-functions-valued Element of bool [:(m /\ (dom ((dom f) --> r))),(C_PFuncs (DOMS T)):]
dom ((dom f) --> r) is set
m /\ (dom ((dom f) --> r)) is set
[:(m /\ (dom ((dom f) --> r))),(C_PFuncs (DOMS T)):] is Relation-like set
bool [:(m /\ (dom ((dom f) --> r))),(C_PFuncs (DOMS T)):] is set
dom (f [#] r) is Element of bool m
dom (f <#> ((dom f) --> r)) is Element of bool (m /\ (dom ((dom f) --> r)))
bool (m /\ (dom ((dom f) --> r))) is set
dom ((dom f) --> r) is Element of bool (dom f)
bool (dom f) is set
(dom f) /\ (dom ((dom f) --> r)) is Element of bool (dom f)
p is set
(f [#] r) . p is Relation-like Function-like complex-valued set
f . p is Relation-like Function-like complex-valued set
r (#) (f . p) is Relation-like Function-like complex-valued set
((dom f) --> r) . p is complex set
(((dom f) --> r) . p) (#) (f . p) is Relation-like Function-like complex-valued set
(f <#> ((dom f) --> r)) . p is Relation-like Function-like complex-valued set
m is set
r is complex set
T is functional complex-functions-membered set
[:m,T:] is Relation-like set
bool [:m,T:] is set
f is Relation-like m -defined T -valued Function-like Function-yielding V156() complex-functions-valued Element of bool [:m,T:]
f [/] r is Relation-like m -defined C_PFuncs (DOMS T) -valued Function-like Function-yielding V156() complex-functions-valued Element of bool [:m,(C_PFuncs (DOMS T)):]
DOMS T is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of T : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of T : verum } is set
C_PFuncs (DOMS T) is functional complex-functions-membered set
[:m,(C_PFuncs (DOMS T)):] is Relation-like set
bool [:m,(C_PFuncs (DOMS T)):] is set
r " is complex set
f [#] (r ") is Relation-like Function-like set
dom f is Element of bool m
bool m is set
(dom f) --> r is Relation-like dom f -defined {r} -valued Function-like constant total V18( dom f,{r}) complex-valued Element of bool [:(dom f),{r}:]
{r} is non empty trivial finite 1 -element complex-membered set
[:(dom f),{r}:] is Relation-like complex-valued set
bool [:(dom f),{r}:] is set
f </> ((dom f) --> r) is Relation-like m /\ (dom ((dom f) --> r)) -defined C_PFuncs (DOMS T) -valued Function-like Function-yielding V156() complex-functions-valued Element of bool [:(m /\ (dom ((dom f) --> r))),(C_PFuncs (DOMS T)):]
dom ((dom f) --> r) is set
m /\ (dom ((dom f) --> r)) is set
[:(m /\ (dom ((dom f) --> r))),(C_PFuncs (DOMS T)):] is Relation-like set
bool [:(m /\ (dom ((dom f) --> r))),(C_PFuncs (DOMS T)):] is set
((dom f) --> r) " is Relation-like dom f -defined Function-like total complex-valued set
f <#> (((dom f) --> r) ") is Relation-like Function-like set
dom (f [/] r) is Element of bool m
dom (f </> ((dom f) --> r)) is Element of bool (m /\ (dom ((dom f) --> r)))
bool (m /\ (dom ((dom f) --> r))) is set
dom ((dom f) --> r) is Element of bool (dom f)
bool (dom f) is set
(dom f) /\ (dom ((dom f) --> r)) is Element of bool (dom f)
p is set
(f [/] r) . p is Relation-like Function-like complex-valued set
f . p is Relation-like Function-like complex-valued set
(f . p) (/) r is Relation-like Function-like complex-valued set
1 / r is complex set
1 * (r ") is complex set
(1 / r) (#) (f . p) is Relation-like Function-like complex-valued set
((dom f) --> r) . p is complex set
(f . p) (/) (((dom f) --> r) . p) is Relation-like Function-like complex-valued set
1 / (((dom f) --> r) . p) is complex set
(((dom f) --> r) . p) " is complex set
1 * ((((dom f) --> r) . p) ") is complex set
(1 / (((dom f) --> r) . p)) (#) (f . p) is Relation-like Function-like complex-valued set
(f </> ((dom f) --> r)) . p is Relation-like Function-like complex-valued set
m is functional complex-functions-membered set
r is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V156() complex-functions-valued V268() FinSequence of m
T is Relation-like NAT -defined m -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V156() complex-functions-valued V268() FinSequence of m
r <++> T is Relation-like NAT /\ NAT -defined C_PFuncs ((DOMS m) /\ (DOMS m)) -valued Function-like Function-yielding V156() complex-functions-valued Element of bool [:(NAT /\ NAT),(C_PFuncs ((DOMS m) /\ (DOMS m))):]
NAT /\ NAT is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
DOMS m is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of m : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of m : verum } is set
(DOMS m) /\ (DOMS m) is set
C_PFuncs ((DOMS m) /\ (DOMS m)) is functional complex-functions-membered set
[:(NAT /\ NAT),(C_PFuncs ((DOMS m) /\ (DOMS m))):] is Relation-like set
bool [:(NAT /\ NAT),(C_PFuncs ((DOMS m) /\ (DOMS m))):] is set
dom (r <++> T) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool (NAT /\ NAT)
bool (NAT /\ NAT) is set
dom r is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom T is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
(dom r) /\ (dom T) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
r <--> T is Relation-like NAT /\ NAT -defined C_PFuncs ((DOMS m) /\ (DOMS m)) -valued Function-like Function-yielding V156() complex-functions-valued Element of bool [:(NAT /\ NAT),(C_PFuncs ((DOMS m) /\ (DOMS m))):]
dom (r <--> T) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool (NAT /\ NAT)
bool (NAT /\ NAT) is set
dom r is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom T is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
(dom r) /\ (dom T) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
r <##> T is Relation-like NAT /\ NAT -defined C_PFuncs ((DOMS m) /\ (DOMS m)) -valued Function-like Function-yielding V156() complex-functions-valued Element of bool [:(NAT /\ NAT),(C_PFuncs ((DOMS m) /\ (DOMS m))):]
dom (r <##> T) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool (NAT /\ NAT)
bool (NAT /\ NAT) is set
dom r is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom T is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
(dom r) /\ (dom T) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
r <//> T is Relation-like NAT /\ NAT -defined C_PFuncs ((DOMS m) /\ (DOMS m)) -valued Function-like Function-yielding V156() complex-functions-valued Element of bool [:(NAT /\ NAT),(C_PFuncs ((DOMS m) /\ (DOMS m))):]
dom (r <//> T) is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool (NAT /\ NAT)
bool (NAT /\ NAT) is set
dom r is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom T is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
(dom r) /\ (dom T) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
m is set
r is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL r is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[:m, the carrier of (TOP-REAL r):] is Relation-like set
bool [:m, the carrier of (TOP-REAL r):] is set
T is Relation-like m -defined the carrier of (TOP-REAL r) -valued Function-like total V18(m, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:m, the carrier of (TOP-REAL r):]
<-> T is Relation-like m -defined R_PFuncs (DOMS the carrier of (TOP-REAL r)) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:m,(R_PFuncs (DOMS the carrier of (TOP-REAL r))):]
DOMS the carrier of (TOP-REAL r) is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
R_PFuncs (DOMS the carrier of (TOP-REAL r)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:m,(R_PFuncs (DOMS the carrier of (TOP-REAL r))):] is Relation-like set
bool [:m,(R_PFuncs (DOMS the carrier of (TOP-REAL r))):] is set
dom (<-> T) is Element of bool m
bool m is set
dom T is Element of bool m
g is set
(<-> T) . g is Relation-like Function-like complex-valued ext-real-valued real-valued set
p is non empty set
[:p, the carrier of (TOP-REAL r):] is Relation-like set
bool [:p, the carrier of (TOP-REAL r):] is set
q is Relation-like p -defined the carrier of (TOP-REAL r) -valued Function-like non empty total V18(p, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:p, the carrier of (TOP-REAL r):]
r is Element of p
q . r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
- (q . r) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) (q . r) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
K260((- 1)) is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K857() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined Function-like constant non empty total complex-valued ext-real-valued real-valued V315() V316() set
{(- 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(dom (id REAL)),{(- 1)}:] is Relation-like complex-valued ext-real-valued real-valued set
<:((dom (id REAL)) --> (- 1)),(id REAL):> is Relation-like Function-like set
<:((dom (id REAL)) --> (- 1)),(id REAL):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(q . r) (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
K853() is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
(q . r) (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
- (q . r) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
- (q . r) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(<-> T) . r is Relation-like Function-like complex-valued ext-real-valued real-valued set
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
r is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL r is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (TOP-REAL m), the carrier of (TOP-REAL r):] is Relation-like set
bool [: the carrier of (TOP-REAL m), the carrier of (TOP-REAL r):] is set
f is Relation-like the carrier of (TOP-REAL m) -defined the carrier of (TOP-REAL r) -valued Function-like non empty total V18( the carrier of (TOP-REAL m), the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL m), the carrier of (TOP-REAL r):]
f (-) is Relation-like Function-like set
dom (f (-)) is set
dom f is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL m)
bool the carrier of (TOP-REAL m) is set
p is set
(f (-)) . p is set
r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) r is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K260((- 1)) is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K857() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined Function-like constant non empty total complex-valued ext-real-valued real-valued V315() V316() set
{(- 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(dom (id REAL)),{(- 1)}:] is Relation-like complex-valued ext-real-valued real-valued set
<:((dom (id REAL)) --> (- 1)),(id REAL):> is Relation-like Function-like set
<:((dom (id REAL)) --> (- 1)),(id REAL):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
r (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
K853() is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
r (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
(f (-)) . r is set
W is Relation-like the carrier of (TOP-REAL m) -defined the carrier of (TOP-REAL r) -valued Function-like non empty total V18( the carrier of (TOP-REAL m), the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL m), the carrier of (TOP-REAL r):]
q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
W . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
m is set
r is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL r is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[:m, the carrier of (TOP-REAL r):] is Relation-like set
bool [:m, the carrier of (TOP-REAL r):] is set
T is ext-real complex real set
f is Relation-like m -defined the carrier of (TOP-REAL r) -valued Function-like total V18(m, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:m, the carrier of (TOP-REAL r):]
f [+] T is Relation-like m -defined R_PFuncs (DOMS the carrier of (TOP-REAL r)) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:m,(R_PFuncs (DOMS the carrier of (TOP-REAL r))):]
DOMS the carrier of (TOP-REAL r) is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
R_PFuncs (DOMS the carrier of (TOP-REAL r)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:m,(R_PFuncs (DOMS the carrier of (TOP-REAL r))):] is Relation-like set
bool [:m,(R_PFuncs (DOMS the carrier of (TOP-REAL r))):] is set
dom f is Element of bool m
bool m is set
dom (f [+] T) is Element of bool m
p is set
(f [+] T) . p is Relation-like Function-like complex-valued ext-real-valued real-valued set
r is non empty set
[:r, the carrier of (TOP-REAL r):] is Relation-like set
bool [:r, the carrier of (TOP-REAL r):] is set
W is Relation-like r -defined the carrier of (TOP-REAL r) -valued Function-like non empty total V18(r, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:r, the carrier of (TOP-REAL r):]
q is Element of r
W . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
T + (W . q) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
(r,(W . q),T) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
(f [+] T) . q is Relation-like Function-like complex-valued ext-real-valued real-valued set
m is set
r is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL r is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[:m, the carrier of (TOP-REAL r):] is Relation-like set
bool [:m, the carrier of (TOP-REAL r):] is set
f is Relation-like m -defined the carrier of (TOP-REAL r) -valued Function-like total V18(m, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:m, the carrier of (TOP-REAL r):]
T is ext-real complex real set
f [-] T is Relation-like m -defined R_PFuncs (DOMS the carrier of (TOP-REAL r)) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:m,(R_PFuncs (DOMS the carrier of (TOP-REAL r))):]
DOMS the carrier of (TOP-REAL r) is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
R_PFuncs (DOMS the carrier of (TOP-REAL r)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:m,(R_PFuncs (DOMS the carrier of (TOP-REAL r))):] is Relation-like set
bool [:m,(R_PFuncs (DOMS the carrier of (TOP-REAL r))):] is set
- T is ext-real complex real set
f [+] (- T) is Relation-like Function-like set
m is set
r is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL r is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[:m, the carrier of (TOP-REAL r):] is Relation-like set
bool [:m, the carrier of (TOP-REAL r):] is set
T is ext-real complex real set
f is Relation-like m -defined the carrier of (TOP-REAL r) -valued Function-like total V18(m, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:m, the carrier of (TOP-REAL r):]
f [#] T is Relation-like m -defined R_PFuncs (DOMS the carrier of (TOP-REAL r)) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:m,(R_PFuncs (DOMS the carrier of (TOP-REAL r))):]
DOMS the carrier of (TOP-REAL r) is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
R_PFuncs (DOMS the carrier of (TOP-REAL r)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:m,(R_PFuncs (DOMS the carrier of (TOP-REAL r))):] is Relation-like set
bool [:m,(R_PFuncs (DOMS the carrier of (TOP-REAL r))):] is set
dom f is Element of bool m
bool m is set
dom (f [#] T) is Element of bool m
p is set
(f [#] T) . p is Relation-like Function-like complex-valued ext-real-valued real-valued set
r is non empty set
[:r, the carrier of (TOP-REAL r):] is Relation-like set
bool [:r, the carrier of (TOP-REAL r):] is set
W is Relation-like r -defined the carrier of (TOP-REAL r) -valued Function-like non empty total V18(r, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:r, the carrier of (TOP-REAL r):]
q is Element of r
W . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
T (#) (W . q) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
K260(T) is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K857() [;] (T,(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> T is Relation-like dom (id REAL) -defined Function-like constant non empty total complex-valued ext-real-valued real-valued set
{T} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(dom (id REAL)),{T}:] is Relation-like complex-valued ext-real-valued real-valued set
<:((dom (id REAL)) --> T),(id REAL):> is Relation-like Function-like set
<:((dom (id REAL)) --> T),(id REAL):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(W . q) (#) K260(T) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
(r,(W . q),T) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
(f [#] T) . q is Relation-like Function-like complex-valued ext-real-valued real-valued set
m is set
r is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL r is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[:m, the carrier of (TOP-REAL r):] is Relation-like set
bool [:m, the carrier of (TOP-REAL r):] is set
f is Relation-like m -defined the carrier of (TOP-REAL r) -valued Function-like total V18(m, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:m, the carrier of (TOP-REAL r):]
T is ext-real complex real set
f [/] T is Relation-like m -defined R_PFuncs (DOMS the carrier of (TOP-REAL r)) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:m,(R_PFuncs (DOMS the carrier of (TOP-REAL r))):]
DOMS the carrier of (TOP-REAL r) is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
R_PFuncs (DOMS the carrier of (TOP-REAL r)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:m,(R_PFuncs (DOMS the carrier of (TOP-REAL r))):] is Relation-like set
bool [:m,(R_PFuncs (DOMS the carrier of (TOP-REAL r))):] is set
T " is ext-real complex real set
f [#] (T ") is Relation-like Function-like set
m is set
r is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL r is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[:m, the carrier of (TOP-REAL r):] is Relation-like set
bool [:m, the carrier of (TOP-REAL r):] is set
T is Relation-like m -defined the carrier of (TOP-REAL r) -valued Function-like total V18(m, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:m, the carrier of (TOP-REAL r):]
f is Relation-like m -defined the carrier of (TOP-REAL r) -valued Function-like total V18(m, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:m, the carrier of (TOP-REAL r):]
T <++> f is Relation-like m /\ m -defined R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r))) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:(m /\ m),(R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r)))):]
m /\ m is set
DOMS the carrier of (TOP-REAL r) is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
(DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r)) is set
R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r))) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:(m /\ m),(R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r)))):] is Relation-like set
bool [:(m /\ m),(R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r)))):] is set
dom T is Element of bool m
bool m is set
dom f is Element of bool m
dom (T <++> f) is Element of bool (m /\ m)
bool (m /\ m) is set
p is set
(T <++> f) . p is Relation-like Function-like complex-valued ext-real-valued real-valued set
r is non empty set
[:r, the carrier of (TOP-REAL r):] is Relation-like set
bool [:r, the carrier of (TOP-REAL r):] is set
W is Relation-like r -defined the carrier of (TOP-REAL r) -valued Function-like non empty total V18(r, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:r, the carrier of (TOP-REAL r):]
q is Element of r
W . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
W1 is Relation-like r -defined the carrier of (TOP-REAL r) -valued Function-like non empty total V18(r, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:r, the carrier of (TOP-REAL r):]
W1 . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
(W . q) + (W1 . q) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
K855() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K855() .: ((W . q),(W1 . q)) is Relation-like Function-like set
<:(W . q),(W1 . q):> is Relation-like Function-like set
<:(W . q),(W1 . q):> (#) K855() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(W . q) + (W1 . q) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
(W . q) + (W1 . q) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(T <++> f) . q is Relation-like Function-like complex-valued ext-real-valued real-valued set
m is set
r is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL r is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[:m, the carrier of (TOP-REAL r):] is Relation-like set
bool [:m, the carrier of (TOP-REAL r):] is set
T is Relation-like m -defined the carrier of (TOP-REAL r) -valued Function-like total V18(m, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:m, the carrier of (TOP-REAL r):]
f is Relation-like m -defined the carrier of (TOP-REAL r) -valued Function-like total V18(m, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:m, the carrier of (TOP-REAL r):]
T <--> f is Relation-like m /\ m -defined R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r))) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:(m /\ m),(R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r)))):]
m /\ m is set
DOMS the carrier of (TOP-REAL r) is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
(DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r)) is set
R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r))) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:(m /\ m),(R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r)))):] is Relation-like set
bool [:(m /\ m),(R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r)))):] is set
dom T is Element of bool m
bool m is set
dom f is Element of bool m
dom (T <--> f) is Element of bool (m /\ m)
bool (m /\ m) is set
p is set
(T <--> f) . p is Relation-like Function-like complex-valued ext-real-valued real-valued set
r is non empty set
[:r, the carrier of (TOP-REAL r):] is Relation-like set
bool [:r, the carrier of (TOP-REAL r):] is set
W is Relation-like r -defined the carrier of (TOP-REAL r) -valued Function-like non empty total V18(r, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:r, the carrier of (TOP-REAL r):]
q is Element of r
W . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
W1 is Relation-like r -defined the carrier of (TOP-REAL r) -valued Function-like non empty total V18(r, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:r, the carrier of (TOP-REAL r):]
W1 . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
(W . q) - (W1 . q) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
- (W1 . q) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) (W1 . q) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
K260((- 1)) is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K857() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined Function-like constant non empty total complex-valued ext-real-valued real-valued V315() V316() set
{(- 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(dom (id REAL)),{(- 1)}:] is Relation-like complex-valued ext-real-valued real-valued set
<:((dom (id REAL)) --> (- 1)),(id REAL):> is Relation-like Function-like set
<:((dom (id REAL)) --> (- 1)),(id REAL):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(W1 . q) (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
K853() is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
(W1 . q) (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
(W . q) + (- (W1 . q)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
K855() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K855() .: ((W . q),(- (W1 . q))) is Relation-like Function-like set
<:(W . q),(- (W1 . q)):> is Relation-like Function-like set
<:(W . q),(- (W1 . q)):> (#) K855() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
K856() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K342(REAL,K855(),(id REAL),K853()) is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K856() .: ((W . q),(W1 . q)) is Relation-like Function-like set
<:(W . q),(W1 . q):> is Relation-like Function-like set
<:(W . q),(W1 . q):> (#) K856() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(W . q) - (W1 . q) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
- (W1 . q) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
- (W1 . q) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(W . q) + (- (W1 . q)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
(W . q) + (- (W1 . q)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() .: ((W . q),(- (W1 . q))) is Relation-like Function-like set
(W . q) - (W1 . q) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(T <--> f) . q is Relation-like Function-like complex-valued ext-real-valued real-valued set
m is set
r is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL r is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[:m, the carrier of (TOP-REAL r):] is Relation-like set
bool [:m, the carrier of (TOP-REAL r):] is set
T is Relation-like m -defined the carrier of (TOP-REAL r) -valued Function-like total V18(m, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:m, the carrier of (TOP-REAL r):]
f is Relation-like m -defined the carrier of (TOP-REAL r) -valued Function-like total V18(m, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:m, the carrier of (TOP-REAL r):]
T <##> f is Relation-like m /\ m -defined R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r))) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:(m /\ m),(R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r)))):]
m /\ m is set
DOMS the carrier of (TOP-REAL r) is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
(DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r)) is set
R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r))) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:(m /\ m),(R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r)))):] is Relation-like set
bool [:(m /\ m),(R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r)))):] is set
dom T is Element of bool m
bool m is set
dom f is Element of bool m
dom (T <##> f) is Element of bool (m /\ m)
bool (m /\ m) is set
p is set
(T <##> f) . p is Relation-like Function-like complex-valued ext-real-valued real-valued set
r is non empty set
[:r, the carrier of (TOP-REAL r):] is Relation-like set
bool [:r, the carrier of (TOP-REAL r):] is set
q is Element of r
(T <##> f) . q is Relation-like Function-like complex-valued ext-real-valued real-valued set
W is Relation-like r -defined the carrier of (TOP-REAL r) -valued Function-like non empty total V18(r, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:r, the carrier of (TOP-REAL r):]
W . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
W1 is Relation-like r -defined the carrier of (TOP-REAL r) -valued Function-like non empty total V18(r, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:r, the carrier of (TOP-REAL r):]
W1 . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
(r,(W . q),(W1 . q)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
K857() .: ((W . q),(W1 . q)) is Relation-like Function-like set
<:(W . q),(W1 . q):> is Relation-like Function-like set
<:(W . q),(W1 . q):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
m is set
r is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL r is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[:m, the carrier of (TOP-REAL r):] is Relation-like set
bool [:m, the carrier of (TOP-REAL r):] is set
T is Relation-like m -defined the carrier of (TOP-REAL r) -valued Function-like total V18(m, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:m, the carrier of (TOP-REAL r):]
f is Relation-like m -defined the carrier of (TOP-REAL r) -valued Function-like total V18(m, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:m, the carrier of (TOP-REAL r):]
T <//> f is Relation-like m /\ m -defined R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r))) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:(m /\ m),(R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r)))):]
m /\ m is set
DOMS the carrier of (TOP-REAL r) is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
(DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r)) is set
R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r))) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:(m /\ m),(R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r)))):] is Relation-like set
bool [:(m /\ m),(R_PFuncs ((DOMS the carrier of (TOP-REAL r)) /\ (DOMS the carrier of (TOP-REAL r)))):] is set
dom T is Element of bool m
bool m is set
dom f is Element of bool m
dom (T <//> f) is Element of bool (m /\ m)
bool (m /\ m) is set
p is set
(T <//> f) . p is Relation-like Function-like complex-valued ext-real-valued real-valued set
r is non empty set
[:r, the carrier of (TOP-REAL r):] is Relation-like set
bool [:r, the carrier of (TOP-REAL r):] is set
q is Element of r
(T <//> f) . q is Relation-like Function-like complex-valued ext-real-valued real-valued set
W is Relation-like r -defined the carrier of (TOP-REAL r) -valued Function-like non empty total V18(r, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:r, the carrier of (TOP-REAL r):]
W . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
W1 is Relation-like r -defined the carrier of (TOP-REAL r) -valued Function-like non empty total V18(r, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:r, the carrier of (TOP-REAL r):]
W1 . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
(r,(W . q),(W1 . q)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
(W1 . q) " is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
(W . q) (#) ((W1 . q) ") is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
K857() .: ((W . q),((W1 . q) ")) is Relation-like Function-like set
<:(W . q),((W1 . q) "):> is Relation-like Function-like set
<:(W . q),((W1 . q) "):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
m is set
[:m, the carrier of R^1:] is Relation-like complex-valued ext-real-valued real-valued set
bool [:m, the carrier of R^1:] is set
r is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL r is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[:m, the carrier of (TOP-REAL r):] is Relation-like set
bool [:m, the carrier of (TOP-REAL r):] is set
T is Relation-like m -defined the carrier of (TOP-REAL r) -valued Function-like total V18(m, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:m, the carrier of (TOP-REAL r):]
f is Relation-like m -defined the carrier of R^1 -valued Function-like total V18(m, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [:m, the carrier of R^1:]
T <+> f is Relation-like m /\ (dom f) -defined R_PFuncs (DOMS the carrier of (TOP-REAL r)) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:(m /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL r))):]
dom f is set
m /\ (dom f) is set
DOMS the carrier of (TOP-REAL r) is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
R_PFuncs (DOMS the carrier of (TOP-REAL r)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:(m /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL r))):] is Relation-like set
bool [:(m /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL r))):] is set
dom T is Element of bool m
bool m is set
dom f is Element of bool m
dom (T <+> f) is Element of bool (m /\ (dom f))
bool (m /\ (dom f)) is set
p is set
(T <+> f) . p is Relation-like Function-like complex-valued ext-real-valued real-valued set
r is non empty set
[:r, the carrier of (TOP-REAL r):] is Relation-like set
bool [:r, the carrier of (TOP-REAL r):] is set
q is Element of r
(T <+> f) . q is Relation-like Function-like complex-valued ext-real-valued real-valued set
W is Relation-like r -defined the carrier of (TOP-REAL r) -valued Function-like non empty total V18(r, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:r, the carrier of (TOP-REAL r):]
W . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
f . q is ext-real complex real set
(r,(W . q),(f . q)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
m is set
[:m, the carrier of R^1:] is Relation-like complex-valued ext-real-valued real-valued set
bool [:m, the carrier of R^1:] is set
r is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL r is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[:m, the carrier of (TOP-REAL r):] is Relation-like set
bool [:m, the carrier of (TOP-REAL r):] is set
T is Relation-like m -defined the carrier of (TOP-REAL r) -valued Function-like total V18(m, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:m, the carrier of (TOP-REAL r):]
f is Relation-like m -defined the carrier of R^1 -valued Function-like total V18(m, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [:m, the carrier of R^1:]
T <-> f is Relation-like m /\ (dom f) -defined R_PFuncs (DOMS the carrier of (TOP-REAL r)) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:(m /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL r))):]
dom f is set
m /\ (dom f) is set
DOMS the carrier of (TOP-REAL r) is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
R_PFuncs (DOMS the carrier of (TOP-REAL r)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:(m /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL r))):] is Relation-like set
bool [:(m /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL r))):] is set
- f is Relation-like m -defined Function-like total complex-valued ext-real-valued real-valued set
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) f is Relation-like m -defined Function-like total complex-valued ext-real-valued real-valued set
T <+> (- f) is Relation-like Function-like set
dom T is Element of bool m
bool m is set
dom f is Element of bool m
dom (T <-> f) is Element of bool (m /\ (dom f))
bool (m /\ (dom f)) is set
p is set
(T <-> f) . p is Relation-like Function-like complex-valued ext-real-valued real-valued set
r is non empty set
[:r, the carrier of (TOP-REAL r):] is Relation-like set
bool [:r, the carrier of (TOP-REAL r):] is set
q is Element of r
(T <-> f) . q is Relation-like Function-like complex-valued ext-real-valued real-valued set
W is Relation-like r -defined the carrier of (TOP-REAL r) -valued Function-like non empty total V18(r, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:r, the carrier of (TOP-REAL r):]
W . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
f . q is ext-real complex real set
(r,(W . q),(f . q)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
- (f . q) is ext-real complex real set
(- (f . q)) + (W . q) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
m is set
[:m, the carrier of R^1:] is Relation-like complex-valued ext-real-valued real-valued set
bool [:m, the carrier of R^1:] is set
r is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL r is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[:m, the carrier of (TOP-REAL r):] is Relation-like set
bool [:m, the carrier of (TOP-REAL r):] is set
T is Relation-like m -defined the carrier of (TOP-REAL r) -valued Function-like total V18(m, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:m, the carrier of (TOP-REAL r):]
f is Relation-like m -defined the carrier of R^1 -valued Function-like total V18(m, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [:m, the carrier of R^1:]
T <#> f is Relation-like m /\ (dom f) -defined R_PFuncs (DOMS the carrier of (TOP-REAL r)) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:(m /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL r))):]
dom f is set
m /\ (dom f) is set
DOMS the carrier of (TOP-REAL r) is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
R_PFuncs (DOMS the carrier of (TOP-REAL r)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:(m /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL r))):] is Relation-like set
bool [:(m /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL r))):] is set
dom T is Element of bool m
bool m is set
dom f is Element of bool m
dom (T <#> f) is Element of bool (m /\ (dom f))
bool (m /\ (dom f)) is set
p is set
(T <#> f) . p is Relation-like Function-like complex-valued ext-real-valued real-valued set
r is non empty set
[:r, the carrier of (TOP-REAL r):] is Relation-like set
bool [:r, the carrier of (TOP-REAL r):] is set
q is Element of r
(T <#> f) . q is Relation-like Function-like complex-valued ext-real-valued real-valued set
W is Relation-like r -defined the carrier of (TOP-REAL r) -valued Function-like non empty total V18(r, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:r, the carrier of (TOP-REAL r):]
W . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
f . q is ext-real complex real set
(r,(W . q),(f . q)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
K260((f . q)) is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K857() [;] ((f . q),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (f . q) is Relation-like dom (id REAL) -defined Function-like constant non empty total complex-valued ext-real-valued real-valued set
{(f . q)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(dom (id REAL)),{(f . q)}:] is Relation-like complex-valued ext-real-valued real-valued set
<:((dom (id REAL)) --> (f . q)),(id REAL):> is Relation-like Function-like set
<:((dom (id REAL)) --> (f . q)),(id REAL):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(W . q) (#) K260((f . q)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
m is set
[:m, the carrier of R^1:] is Relation-like complex-valued ext-real-valued real-valued set
bool [:m, the carrier of R^1:] is set
r is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL r is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[:m, the carrier of (TOP-REAL r):] is Relation-like set
bool [:m, the carrier of (TOP-REAL r):] is set
T is Relation-like m -defined the carrier of (TOP-REAL r) -valued Function-like total V18(m, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:m, the carrier of (TOP-REAL r):]
f is Relation-like m -defined the carrier of R^1 -valued Function-like total V18(m, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [:m, the carrier of R^1:]
T </> f is Relation-like m /\ (dom f) -defined R_PFuncs (DOMS the carrier of (TOP-REAL r)) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:(m /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL r))):]
dom f is set
m /\ (dom f) is set
DOMS the carrier of (TOP-REAL r) is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL r) : verum } is set
R_PFuncs (DOMS the carrier of (TOP-REAL r)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:(m /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL r))):] is Relation-like set
bool [:(m /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL r))):] is set
f " is Relation-like m -defined Function-like total complex-valued ext-real-valued real-valued set
T <#> (f ") is Relation-like Function-like set
dom T is Element of bool m
bool m is set
dom f is Element of bool m
dom (T </> f) is Element of bool (m /\ (dom f))
bool (m /\ (dom f)) is set
p is set
(T </> f) . p is Relation-like Function-like complex-valued ext-real-valued real-valued set
r is non empty set
[:r, the carrier of (TOP-REAL r):] is Relation-like set
bool [:r, the carrier of (TOP-REAL r):] is set
q is Element of r
(T </> f) . q is Relation-like Function-like complex-valued ext-real-valued real-valued set
W is Relation-like r -defined the carrier of (TOP-REAL r) -valued Function-like non empty total V18(r, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:r, the carrier of (TOP-REAL r):]
W . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
f . q is ext-real complex real set
(r,(W . q),(f . q)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
1 / (f . q) is ext-real complex real set
(f . q) " is ext-real complex real set
1 * ((f . q) ") is ext-real complex real set
(1 / (f . q)) (#) (W . q) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
K260((1 / (f . q))) is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K857() [;] ((1 / (f . q)),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (1 / (f . q)) is Relation-like dom (id REAL) -defined Function-like constant non empty total complex-valued ext-real-valued real-valued set
{(1 / (f . q))} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(dom (id REAL)),{(1 / (f . q))}:] is Relation-like complex-valued ext-real-valued real-valued set
<:((dom (id REAL)) --> (1 / (f . q))),(id REAL):> is Relation-like Function-like set
<:((dom (id REAL)) --> (1 / (f . q))),(id REAL):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(W . q) (#) K260((1 / (f . q))) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
r is non empty set
T is complex-membered ext-real-membered real-membered set
[:r,T:] is Relation-like complex-valued ext-real-valued real-valued set
bool [:r,T:] is set
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[:r, the carrier of (TOP-REAL m):] is Relation-like set
bool [:r, the carrier of (TOP-REAL m):] is set
f is Relation-like r -defined T -valued Function-like V18(r,T) complex-valued ext-real-valued real-valued Element of bool [:r,T:]
g is Element of r
f . g is ext-real complex real set
m |-> (f . g) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
Seg m is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
(Seg m) --> (f . g) is Relation-like Seg m -defined Seg m -defined {(f . g)} -valued Function-like constant total total V18( Seg m,{(f . g)}) complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() Element of bool [:(Seg m),{(f . g)}:]
{(f . g)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(Seg m),{(f . g)}:] is Relation-like complex-valued ext-real-valued real-valued finite set
bool [:(Seg m),{(f . g)}:] is finite V49() set
f . g is ext-real complex real Element of T
m |-> (f . g) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
(Seg m) --> (f . g) is Relation-like Seg m -defined Seg m -defined {(f . g)} -valued Function-like constant total total V18( Seg m,{(f . g)}) complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() Element of bool [:(Seg m),{(f . g)}:]
{(f . g)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(Seg m),{(f . g)}:] is Relation-like complex-valued ext-real-valued real-valued finite set
bool [:(Seg m),{(f . g)}:] is finite V49() set
REAL m is functional non empty FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V268() Element of REAL * : len b₁ = m } is set
p is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
g is Relation-like r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18(r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:r, the carrier of (TOP-REAL m):]
g is Relation-like r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18(r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:r, the carrier of (TOP-REAL m):]
p is Relation-like r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18(r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:r, the carrier of (TOP-REAL m):]
r is Element of r
g . r is Relation-like Function-like complex-valued ext-real-valued real-valued set
p . r is Relation-like Function-like complex-valued ext-real-valued real-valued set
g . r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
f . r is ext-real complex real Element of T
m |-> (f . r) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
Seg m is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
(Seg m) --> (f . r) is Relation-like Seg m -defined Seg m -defined {(f . r)} -valued Function-like constant total total V18( Seg m,{(f . r)}) complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() Element of bool [:(Seg m),{(f . r)}:]
{(f . r)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(Seg m),{(f . r)}:] is Relation-like complex-valued ext-real-valued real-valued finite set
bool [:(Seg m),{(f . r)}:] is finite V49() set
p . r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
m is set
r is ext-real non negative ordinal natural complex real finite cardinal set
Seg r is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= r ) } is set
TOP-REAL r is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
T is non empty TopSpace-like TopStruct
the carrier of T is non empty set
f is complex-membered ext-real-membered real-membered set
[: the carrier of T,f:] is Relation-like complex-valued ext-real-valued real-valued set
bool [: the carrier of T,f:] is set
g is Relation-like the carrier of T -defined f -valued Function-like V18( the carrier of T,f) complex-valued ext-real-valued real-valued Element of bool [: the carrier of T,f:]
(r, the carrier of T,f,g) is Relation-like the carrier of T -defined the carrier of (TOP-REAL r) -valued Function-like non empty total V18( the carrier of T, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of T, the carrier of (TOP-REAL r):]
[: the carrier of T, the carrier of (TOP-REAL r):] is Relation-like set
bool [: the carrier of T, the carrier of (TOP-REAL r):] is set
p is Element of the carrier of T
(r, the carrier of T,f,g) . p is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
(r,((r, the carrier of T,f,g) . p),m) is ext-real complex real Element of REAL
g . p is ext-real complex real Element of f
r |-> (g . p) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
(Seg r) --> (g . p) is Relation-like Seg r -defined Seg r -defined {(g . p)} -valued Function-like constant total total V18( Seg r,{(g . p)}) complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() Element of bool [:(Seg r),{(g . p)}:]
{(g . p)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(Seg r),{(g . p)}:] is Relation-like complex-valued ext-real-valued real-valued finite set
bool [:(Seg r),{(g . p)}:] is finite V49() set
(r |-> (g . p)) . m is ext-real complex real set
m is non empty set
m --> 0 is Relation-like non non-empty m -defined REAL -valued RAT -valued INT -valued Function-like constant non empty total V18(m, REAL ) complex-valued ext-real-valued real-valued natural-valued Function-yielding V156() V316() V317() Element of bool [:m,REAL:]
[:m,REAL:] is Relation-like non trivial complex-valued ext-real-valued real-valued non finite set
bool [:m,REAL:] is non trivial non finite set
{0} is functional non empty trivial finite V49() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered complex-functions-membered ext-real-functions-membered real-functions-membered set
[:m,{0}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
r is complex-membered ext-real-membered real-membered set
[:m,r:] is Relation-like complex-valued ext-real-valued real-valued set
bool [:m,r:] is set
T is Relation-like m -defined r -valued Function-like V18(m,r) complex-valued ext-real-valued real-valued Element of bool [:m,r:]
(0,m,r,T) is Relation-like m -defined the carrier of (TOP-REAL 0) -valued Function-like non empty total V18(m, the carrier of (TOP-REAL 0)) complex-valued ext-real-valued real-valued Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:m, the carrier of (TOP-REAL 0):]
[:m, the carrier of (TOP-REAL 0):] is Relation-like complex-valued ext-real-valued real-valued set
bool [:m, the carrier of (TOP-REAL 0):] is set
f is Relation-like non-empty empty-yielding NAT -defined REAL -valued Function-like one-to-one constant functional empty ext-real non positive non negative ordinal natural complex complex-valued ext-real-valued real-valued natural-valued real finite finite-yielding V49() cardinal 0 -element FinSequence-like FinSubsequence-like FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V132() Function-yielding V156() V157() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued V267() V268() Element of the carrier of (TOP-REAL 0)
m --> f is Relation-like m -defined RAT -valued the carrier of (TOP-REAL 0) -valued Function-like constant non empty total V18(m, the carrier of (TOP-REAL 0)) complex-valued ext-real-valued real-valued natural-valued Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued V316() V317() Element of bool [:m, the carrier of (TOP-REAL 0):]
{f} is functional non empty trivial finite V49() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered complex-functions-membered ext-real-functions-membered real-functions-membered set
[:m,{f}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
g is Element of m
(0,m,r,T) . g is Relation-like Function-like ext-real complex complex-valued ext-real-valued real-valued real set
(m --> f) . g is Relation-like Function-like ext-real non negative ordinal natural complex complex-valued ext-real-valued real-valued real V44() finite cardinal V268() set
(0,m,r,T) . g is Relation-like non-empty empty-yielding NAT -defined REAL -valued Function-like one-to-one constant functional empty ext-real non positive non negative ordinal natural complex complex-valued ext-real-valued real-valued natural-valued real finite finite-yielding V49() cardinal 0 -element FinSequence-like FinSubsequence-like FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V132() Function-yielding V156() V157() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued V267() V268() Element of the carrier of (TOP-REAL 0)
(m --> f) . g is Relation-like non-empty empty-yielding NAT -defined REAL -valued Function-like one-to-one constant functional empty ext-real non positive non negative ordinal natural complex complex-valued ext-real-valued real-valued natural-valued real V44() finite finite-yielding V49() cardinal 0 -element FinSequence-like FinSubsequence-like FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V132() Function-yielding V156() V157() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued V267() V268() Element of the carrier of (TOP-REAL 0)
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
r is non empty TopSpace-like TopStruct
the carrier of r is non empty set
[: the carrier of r, the carrier of (TOP-REAL m):] is Relation-like set
bool [: the carrier of r, the carrier of (TOP-REAL m):] is set
[: the carrier of r, the carrier of R^1:] is Relation-like complex-valued ext-real-valued real-valued set
bool [: the carrier of r, the carrier of R^1:] is set
T is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
f is Relation-like the carrier of r -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of r, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of r, the carrier of R^1:]
T <+> f is Relation-like the carrier of r /\ (dom f) -defined R_PFuncs (DOMS the carrier of (TOP-REAL m)) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:( the carrier of r /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL m))):]
dom f is non empty set
the carrier of r /\ (dom f) is set
DOMS the carrier of (TOP-REAL m) is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL m) : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL m) : verum } is set
R_PFuncs (DOMS the carrier of (TOP-REAL m)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:( the carrier of r /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL m))):] is Relation-like set
bool [:( the carrier of r /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL m))):] is set
(m, the carrier of r, the carrier of R^1,f) is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
T <++> (m, the carrier of r, the carrier of R^1,f) is Relation-like the carrier of r /\ the carrier of r -defined R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m))) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):]
the carrier of r /\ the carrier of r is set
(DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)) is set
R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m))) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):] is Relation-like set
bool [:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):] is set
p is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
r is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
q is Element of the carrier of r
p . q is Relation-like Function-like complex-valued ext-real-valued real-valued set
r . q is Relation-like Function-like complex-valued ext-real-valued real-valued set
dom p is non empty Element of bool the carrier of r
bool the carrier of r is set
T . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(m, the carrier of r, the carrier of R^1,f) . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(T . q) + ((m, the carrier of r, the carrier of R^1,f) . q) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(T . q) + ((m, the carrier of r, the carrier of R^1,f) . q) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K855() .: ((T . q),((m, the carrier of r, the carrier of R^1,f) . q)) is Relation-like Function-like set
<:(T . q),((m, the carrier of r, the carrier of R^1,f) . q):> is Relation-like Function-like set
<:(T . q),((m, the carrier of r, the carrier of R^1,f) . q):> (#) K855() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
f . q is ext-real complex real Element of the carrier of R^1
(m,(T . q),(f . q)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
dom (T . q) is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
Seg m is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
dom ((m, the carrier of r, the carrier of R^1,f) . q) is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom ((T . q) + ((m, the carrier of r, the carrier of R^1,f) . q)) is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
(dom (T . q)) /\ (dom ((m, the carrier of r, the carrier of R^1,f) . q)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom (m,(T . q),(f . q)) is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom ((T . q) + ((m, the carrier of r, the carrier of R^1,f) . q)) is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
W is set
((T . q) + ((m, the carrier of r, the carrier of R^1,f) . q)) . W is ext-real complex real set
(m,(T . q),(f . q)) . W is ext-real complex real set
(m,((T . q) + ((m, the carrier of r, the carrier of R^1,f) . q)),W) is ext-real complex real Element of REAL
(m,(T . q),W) is ext-real complex real Element of REAL
(m,((m, the carrier of r, the carrier of R^1,f) . q),W) is ext-real complex real Element of REAL
(m,(T . q),W) + (m,((m, the carrier of r, the carrier of R^1,f) . q),W) is ext-real complex real set
(m,(T . q),W) + (f . q) is ext-real complex real set
(m,(m,(T . q),(f . q)),W) is ext-real complex real Element of REAL
dom r is non empty Element of bool the carrier of r
r . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
p . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
r is non empty TopSpace-like TopStruct
the carrier of r is non empty set
[: the carrier of r, the carrier of (TOP-REAL m):] is Relation-like set
bool [: the carrier of r, the carrier of (TOP-REAL m):] is set
[: the carrier of r, the carrier of R^1:] is Relation-like complex-valued ext-real-valued real-valued set
bool [: the carrier of r, the carrier of R^1:] is set
T is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
f is Relation-like the carrier of r -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of r, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of r, the carrier of R^1:]
T <-> f is Relation-like the carrier of r /\ (dom f) -defined R_PFuncs (DOMS the carrier of (TOP-REAL m)) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:( the carrier of r /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL m))):]
dom f is non empty set
the carrier of r /\ (dom f) is set
DOMS the carrier of (TOP-REAL m) is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL m) : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL m) : verum } is set
R_PFuncs (DOMS the carrier of (TOP-REAL m)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:( the carrier of r /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL m))):] is Relation-like set
bool [:( the carrier of r /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL m))):] is set
- f is Relation-like the carrier of r -defined Function-like total complex-valued ext-real-valued real-valued set
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) f is Relation-like the carrier of r -defined Function-like total complex-valued ext-real-valued real-valued set
T <+> (- f) is Relation-like Function-like set
(m, the carrier of r, the carrier of R^1,f) is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
T <--> (m, the carrier of r, the carrier of R^1,f) is Relation-like the carrier of r /\ the carrier of r -defined R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m))) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):]
the carrier of r /\ the carrier of r is set
(DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)) is set
R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m))) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):] is Relation-like set
bool [:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):] is set
p is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
r is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
q is Element of the carrier of r
p . q is Relation-like Function-like complex-valued ext-real-valued real-valued set
r . q is Relation-like Function-like complex-valued ext-real-valued real-valued set
dom p is non empty Element of bool the carrier of r
bool the carrier of r is set
T . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(m, the carrier of r, the carrier of R^1,f) . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(T . q) - ((m, the carrier of r, the carrier of R^1,f) . q) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- ((m, the carrier of r, the carrier of R^1,f) . q) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- ((m, the carrier of r, the carrier of R^1,f) . q) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(- 1) (#) ((m, the carrier of r, the carrier of R^1,f) . q) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K260((- 1)) is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K857() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined Function-like constant non empty total complex-valued ext-real-valued real-valued V315() V316() set
{(- 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(dom (id REAL)),{(- 1)}:] is Relation-like complex-valued ext-real-valued real-valued set
<:((dom (id REAL)) --> (- 1)),(id REAL):> is Relation-like Function-like set
<:((dom (id REAL)) --> (- 1)),(id REAL):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
((m, the carrier of r, the carrier of R^1,f) . q) (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
K853() is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
((m, the carrier of r, the carrier of R^1,f) . q) (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
(T . q) + (- ((m, the carrier of r, the carrier of R^1,f) . q)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(T . q) + (- ((m, the carrier of r, the carrier of R^1,f) . q)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K855() .: ((T . q),(- ((m, the carrier of r, the carrier of R^1,f) . q))) is Relation-like Function-like set
(T . q) - ((m, the carrier of r, the carrier of R^1,f) . q) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
- ((m, the carrier of r, the carrier of R^1,f) . q) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
(T . q) + (- ((m, the carrier of r, the carrier of R^1,f) . q)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K855() .: ((T . q),(- ((m, the carrier of r, the carrier of R^1,f) . q))) is Relation-like Function-like set
K856() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K342(REAL,K855(),(id REAL),K853()) is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K856() .: ((T . q),((m, the carrier of r, the carrier of R^1,f) . q)) is Relation-like Function-like set
<:(T . q),((m, the carrier of r, the carrier of R^1,f) . q):> is Relation-like Function-like set
<:(T . q),((m, the carrier of r, the carrier of R^1,f) . q):> (#) K856() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
f . q is ext-real complex real Element of the carrier of R^1
(m,(T . q),(f . q)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- (f . q) is ext-real complex real set
(- (f . q)) + (T . q) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
dom (T . q) is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
Seg m is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
dom ((m, the carrier of r, the carrier of R^1,f) . q) is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom ((T . q) - ((m, the carrier of r, the carrier of R^1,f) . q)) is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
(dom (T . q)) /\ (dom ((m, the carrier of r, the carrier of R^1,f) . q)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom (m,(T . q),(f . q)) is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom ((T . q) - ((m, the carrier of r, the carrier of R^1,f) . q)) is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
W is set
((T . q) - ((m, the carrier of r, the carrier of R^1,f) . q)) . W is ext-real complex real set
(m,(T . q),(f . q)) . W is ext-real complex real set
(m,((T . q) - ((m, the carrier of r, the carrier of R^1,f) . q)),W) is ext-real complex real Element of REAL
(m,(T . q),W) is ext-real complex real Element of REAL
(m,((m, the carrier of r, the carrier of R^1,f) . q),W) is ext-real complex real Element of REAL
(m,(T . q),W) - (m,((m, the carrier of r, the carrier of R^1,f) . q),W) is ext-real complex real set
- (m,((m, the carrier of r, the carrier of R^1,f) . q),W) is ext-real complex real set
(m,(T . q),W) + (- (m,((m, the carrier of r, the carrier of R^1,f) . q),W)) is ext-real complex real set
(m,(T . q),W) - (f . q) is ext-real complex real set
(m,(T . q),W) + (- (f . q)) is ext-real complex real set
(m,(m,(T . q),(f . q)),W) is ext-real complex real Element of REAL
dom r is non empty Element of bool the carrier of r
r . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
p . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
r is non empty TopSpace-like TopStruct
the carrier of r is non empty set
[: the carrier of r, the carrier of (TOP-REAL m):] is Relation-like set
bool [: the carrier of r, the carrier of (TOP-REAL m):] is set
[: the carrier of r, the carrier of R^1:] is Relation-like complex-valued ext-real-valued real-valued set
bool [: the carrier of r, the carrier of R^1:] is set
T is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
f is Relation-like the carrier of r -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of r, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of r, the carrier of R^1:]
T <#> f is Relation-like the carrier of r /\ (dom f) -defined R_PFuncs (DOMS the carrier of (TOP-REAL m)) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:( the carrier of r /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL m))):]
dom f is non empty set
the carrier of r /\ (dom f) is set
DOMS the carrier of (TOP-REAL m) is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL m) : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL m) : verum } is set
R_PFuncs (DOMS the carrier of (TOP-REAL m)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:( the carrier of r /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL m))):] is Relation-like set
bool [:( the carrier of r /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL m))):] is set
(m, the carrier of r, the carrier of R^1,f) is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
T <##> (m, the carrier of r, the carrier of R^1,f) is Relation-like the carrier of r /\ the carrier of r -defined R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m))) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):]
the carrier of r /\ the carrier of r is set
(DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)) is set
R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m))) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):] is Relation-like set
bool [:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):] is set
p is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
r is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
q is Element of the carrier of r
p . q is Relation-like Function-like complex-valued ext-real-valued real-valued set
r . q is Relation-like Function-like complex-valued ext-real-valued real-valued set
dom p is non empty Element of bool the carrier of r
bool the carrier of r is set
T . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(m, the carrier of r, the carrier of R^1,f) . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(m,(T . q),((m, the carrier of r, the carrier of R^1,f) . q)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
K857() .: ((T . q),((m, the carrier of r, the carrier of R^1,f) . q)) is Relation-like Function-like set
<:(T . q),((m, the carrier of r, the carrier of R^1,f) . q):> is Relation-like Function-like set
<:(T . q),((m, the carrier of r, the carrier of R^1,f) . q):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
f . q is ext-real complex real Element of the carrier of R^1
(m,(T . q),(f . q)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
K260((f . q)) is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K857() [;] ((f . q),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (f . q) is Relation-like dom (id REAL) -defined the carrier of R^1 -valued Function-like constant non empty total complex-valued ext-real-valued real-valued set
{(f . q)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(dom (id REAL)),{(f . q)}:] is Relation-like complex-valued ext-real-valued real-valued set
<:((dom (id REAL)) --> (f . q)),(id REAL):> is Relation-like Function-like set
<:((dom (id REAL)) --> (f . q)),(id REAL):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(T . q) (#) K260((f . q)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
dom (T . q) is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
Seg m is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
dom ((m, the carrier of r, the carrier of R^1,f) . q) is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom (m,(T . q),((m, the carrier of r, the carrier of R^1,f) . q)) is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
(dom (T . q)) /\ (dom ((m, the carrier of r, the carrier of R^1,f) . q)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom (m,(T . q),(f . q)) is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom (m,(T . q),((m, the carrier of r, the carrier of R^1,f) . q)) is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
W is set
(m,(T . q),((m, the carrier of r, the carrier of R^1,f) . q)) . W is ext-real complex real set
(m,(T . q),(f . q)) . W is ext-real complex real set
(m,(m,(T . q),((m, the carrier of r, the carrier of R^1,f) . q)),W) is ext-real complex real Element of REAL
(m,(T . q),W) is ext-real complex real Element of REAL
(m,((m, the carrier of r, the carrier of R^1,f) . q),W) is ext-real complex real Element of REAL
(m,(T . q),W) * (m,((m, the carrier of r, the carrier of R^1,f) . q),W) is ext-real complex real set
(m,(T . q),W) * (f . q) is ext-real complex real set
(m,(m,(T . q),(f . q)),W) is ext-real complex real Element of REAL
dom r is non empty Element of bool the carrier of r
r . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
p . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
r is non empty TopSpace-like TopStruct
the carrier of r is non empty set
[: the carrier of r, the carrier of (TOP-REAL m):] is Relation-like set
bool [: the carrier of r, the carrier of (TOP-REAL m):] is set
[: the carrier of r, the carrier of R^1:] is Relation-like complex-valued ext-real-valued real-valued set
bool [: the carrier of r, the carrier of R^1:] is set
T is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
f is Relation-like the carrier of r -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of r, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of r, the carrier of R^1:]
T </> f is Relation-like the carrier of r /\ (dom f) -defined R_PFuncs (DOMS the carrier of (TOP-REAL m)) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:( the carrier of r /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL m))):]
dom f is non empty set
the carrier of r /\ (dom f) is set
DOMS the carrier of (TOP-REAL m) is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL m) : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL m) : verum } is set
R_PFuncs (DOMS the carrier of (TOP-REAL m)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:( the carrier of r /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL m))):] is Relation-like set
bool [:( the carrier of r /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL m))):] is set
f " is Relation-like the carrier of r -defined Function-like total complex-valued ext-real-valued real-valued set
T <#> (f ") is Relation-like Function-like set
(m, the carrier of r, the carrier of R^1,f) is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
T <//> (m, the carrier of r, the carrier of R^1,f) is Relation-like the carrier of r /\ the carrier of r -defined R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m))) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):]
the carrier of r /\ the carrier of r is set
(DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)) is set
R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m))) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):] is Relation-like set
bool [:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):] is set
p is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
r is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
q is Element of the carrier of r
p . q is Relation-like Function-like complex-valued ext-real-valued real-valued set
r . q is Relation-like Function-like complex-valued ext-real-valued real-valued set
dom p is non empty Element of bool the carrier of r
bool the carrier of r is set
T . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(m, the carrier of r, the carrier of R^1,f) . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(m,(T . q),((m, the carrier of r, the carrier of R^1,f) . q)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
((m, the carrier of r, the carrier of R^1,f) . q) " is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
(T . q) (#) (((m, the carrier of r, the carrier of R^1,f) . q) ") is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K857() .: ((T . q),(((m, the carrier of r, the carrier of R^1,f) . q) ")) is Relation-like Function-like set
<:(T . q),(((m, the carrier of r, the carrier of R^1,f) . q) "):> is Relation-like Function-like set
<:(T . q),(((m, the carrier of r, the carrier of R^1,f) . q) "):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
f " is Relation-like the carrier of r -defined REAL -valued Function-like non empty total V18( the carrier of r, REAL ) complex-valued ext-real-valued real-valued Element of bool [: the carrier of r,REAL:]
[: the carrier of r,REAL:] is Relation-like non trivial complex-valued ext-real-valued real-valued non finite set
bool [: the carrier of r,REAL:] is non trivial non finite set
(f ") . q is ext-real complex real Element of REAL
(m,(T . q),((f ") . q)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
K260(((f ") . q)) is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K857() [;] (((f ") . q),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> ((f ") . q) is Relation-like dom (id REAL) -defined REAL -valued Function-like constant non empty total complex-valued ext-real-valued real-valued set
{((f ") . q)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(dom (id REAL)),{((f ") . q)}:] is Relation-like complex-valued ext-real-valued real-valued set
<:((dom (id REAL)) --> ((f ") . q)),(id REAL):> is Relation-like Function-like set
<:((dom (id REAL)) --> ((f ") . q)),(id REAL):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(T . q) (#) K260(((f ") . q)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
dom (T . q) is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
Seg m is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
dom ((m, the carrier of r, the carrier of R^1,f) . q) is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom (m,(T . q),((m, the carrier of r, the carrier of R^1,f) . q)) is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
(dom (T . q)) /\ (dom ((m, the carrier of r, the carrier of R^1,f) . q)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom (m,(T . q),((f ") . q)) is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
dom (m,(T . q),((m, the carrier of r, the carrier of R^1,f) . q)) is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
W is set
(m,(T . q),((m, the carrier of r, the carrier of R^1,f) . q)) . W is ext-real complex real set
(m,(T . q),((f ") . q)) . W is ext-real complex real set
(m,(m,(T . q),((m, the carrier of r, the carrier of R^1,f) . q)),W) is ext-real complex real Element of REAL
(m,(T . q),W) is ext-real complex real Element of REAL
(m,((m, the carrier of r, the carrier of R^1,f) . q),W) is ext-real complex real Element of REAL
(m,(T . q),W) / (m,((m, the carrier of r, the carrier of R^1,f) . q),W) is ext-real complex real Element of COMPLEX
(m,((m, the carrier of r, the carrier of R^1,f) . q),W) " is ext-real complex real set
(m,(T . q),W) * ((m,((m, the carrier of r, the carrier of R^1,f) . q),W) ") is ext-real complex real set
f . q is ext-real complex real Element of the carrier of R^1
(m,(T . q),W) / (f . q) is ext-real complex real Element of COMPLEX
(f . q) " is ext-real complex real set
(m,(T . q),W) * ((f . q) ") is ext-real complex real set
(m,(T . q),W) * ((f ") . q) is ext-real complex real set
(m,(m,(T . q),((f ") . q)),W) is ext-real complex real Element of REAL
dom r is non empty Element of bool the carrier of r
r . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
p . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[:(TOP-REAL m),(TOP-REAL m):] is non empty strict TopSpace-like TopStruct
the carrier of [:(TOP-REAL m),(TOP-REAL m):] is non empty set
[: the carrier of (TOP-REAL m), the carrier of (TOP-REAL m):] is Relation-like set
[: the carrier of [:(TOP-REAL m),(TOP-REAL m):], the carrier of (TOP-REAL m):] is Relation-like set
bool [: the carrier of [:(TOP-REAL m),(TOP-REAL m):], the carrier of (TOP-REAL m):] is set
[:[: the carrier of (TOP-REAL m), the carrier of (TOP-REAL m):], the carrier of (TOP-REAL m):] is Relation-like set
bool [:[: the carrier of (TOP-REAL m), the carrier of (TOP-REAL m):], the carrier of (TOP-REAL m):] is set
f is Relation-like [: the carrier of (TOP-REAL m), the carrier of (TOP-REAL m):] -defined the carrier of (TOP-REAL m) -valued Function-like total V18([: the carrier of (TOP-REAL m), the carrier of (TOP-REAL m):], the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:[: the carrier of (TOP-REAL m), the carrier of (TOP-REAL m):], the carrier of (TOP-REAL m):]
f is Relation-like the carrier of [:(TOP-REAL m),(TOP-REAL m):] -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of [:(TOP-REAL m),(TOP-REAL m):], the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of [:(TOP-REAL m),(TOP-REAL m):], the carrier of (TOP-REAL m):]
g is Relation-like the carrier of [:(TOP-REAL m),(TOP-REAL m):] -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of [:(TOP-REAL m),(TOP-REAL m):], the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of [:(TOP-REAL m),(TOP-REAL m):], the carrier of (TOP-REAL m):]
p is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
((TOP-REAL m),(TOP-REAL m),(TOP-REAL m),f,p,r) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
[p,r] is set
{p,r} is functional non empty finite V49() complex-functions-membered ext-real-functions-membered real-functions-membered set
{p} is functional non empty trivial finite V49() 1 -element complex-functions-membered ext-real-functions-membered real-functions-membered set
{{p,r},{p}} is non empty finite V49() set
f . [p,r] is Relation-like Function-like complex-valued ext-real-valued real-valued set
(m,p,r) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
K857() .: (p,r) is Relation-like Function-like set
<:p,r:> is Relation-like Function-like set
<:p,r:> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
((TOP-REAL m),(TOP-REAL m),(TOP-REAL m),g,p,r) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
g . [p,r] is Relation-like Function-like complex-valued ext-real-valued real-valued set
[:(TOP-REAL 0),(TOP-REAL 0):] is non empty strict TopSpace-like TopStruct
the carrier of [:(TOP-REAL 0),(TOP-REAL 0):] is non empty set
(0) is Relation-like the carrier of [:(TOP-REAL 0),(TOP-REAL 0):] -defined the carrier of (TOP-REAL 0) -valued Function-like non empty total V18( the carrier of [:(TOP-REAL 0),(TOP-REAL 0):], the carrier of (TOP-REAL 0)) complex-valued ext-real-valued real-valued Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of [:(TOP-REAL 0),(TOP-REAL 0):], the carrier of (TOP-REAL 0):]
[: the carrier of [:(TOP-REAL 0),(TOP-REAL 0):], the carrier of (TOP-REAL 0):] is Relation-like complex-valued ext-real-valued real-valued set
bool [: the carrier of [:(TOP-REAL 0),(TOP-REAL 0):], the carrier of (TOP-REAL 0):] is set
[:(TOP-REAL 0),(TOP-REAL 0):] --> (0. (TOP-REAL 0)) is Relation-like the carrier of [:(TOP-REAL 0),(TOP-REAL 0):] -defined the carrier of (TOP-REAL 0) -valued Function-like non empty total V18( the carrier of [:(TOP-REAL 0),(TOP-REAL 0):], the carrier of (TOP-REAL 0)) complex-valued ext-real-valued real-valued Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued continuous Element of bool [: the carrier of [:(TOP-REAL 0),(TOP-REAL 0):], the carrier of (TOP-REAL 0):]
the carrier of [:(TOP-REAL 0),(TOP-REAL 0):] --> (0. (TOP-REAL 0)) is Relation-like the carrier of [:(TOP-REAL 0),(TOP-REAL 0):] -defined RAT -valued the carrier of (TOP-REAL 0) -valued the carrier of (TOP-REAL 0) -valued Function-like constant non empty total V18( the carrier of [:(TOP-REAL 0),(TOP-REAL 0):], the carrier of (TOP-REAL 0)) complex-valued ext-real-valued real-valued natural-valued Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued V316() V317() Element of bool [: the carrier of [:(TOP-REAL 0),(TOP-REAL 0):], the carrier of (TOP-REAL 0):]
[: the carrier of [:(TOP-REAL 0),(TOP-REAL 0):],{(0. (TOP-REAL 0))}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
r is Element of the carrier of [:(TOP-REAL 0),(TOP-REAL 0):]
(0) . r is Relation-like Function-like ext-real complex complex-valued ext-real-valued real-valued real set
([:(TOP-REAL 0),(TOP-REAL 0):] --> (0. (TOP-REAL 0))) . r is Relation-like Function-like ext-real complex complex-valued ext-real-valued real-valued real set
((TOP-REAL 0),(TOP-REAL 0),(TOP-REAL 0),(0),r) is Relation-like non-empty empty-yielding NAT -defined REAL -valued Function-like one-to-one constant functional empty ext-real non positive non negative ordinal natural complex complex-valued ext-real-valued real-valued natural-valued real finite finite-yielding V49() cardinal 0 -element FinSequence-like FinSubsequence-like FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V132() Function-yielding V156() V157() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued V267() V268() Element of the carrier of (TOP-REAL 0)
((TOP-REAL 0),(TOP-REAL 0),(TOP-REAL 0),([:(TOP-REAL 0),(TOP-REAL 0):] --> (0. (TOP-REAL 0))),r) is Relation-like non-empty empty-yielding NAT -defined REAL -valued Function-like one-to-one constant functional empty ext-real non positive non negative ordinal natural complex complex-valued ext-real-valued real-valued natural-valued real finite finite-yielding V49() cardinal 0 -element FinSequence-like FinSubsequence-like FinSequence-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V132() Function-yielding V156() V157() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued integer-functions-valued natural-functions-valued V267() V268() Element of the carrier of (TOP-REAL 0)
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
(m) is Relation-like the carrier of [:(TOP-REAL m),(TOP-REAL m):] -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of [:(TOP-REAL m),(TOP-REAL m):], the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of [:(TOP-REAL m),(TOP-REAL m):], the carrier of (TOP-REAL m):]
[:(TOP-REAL m),(TOP-REAL m):] is non empty strict TopSpace-like TopStruct
the carrier of [:(TOP-REAL m),(TOP-REAL m):] is non empty set
[: the carrier of [:(TOP-REAL m),(TOP-REAL m):], the carrier of (TOP-REAL m):] is Relation-like set
bool [: the carrier of [:(TOP-REAL m),(TOP-REAL m):], the carrier of (TOP-REAL m):] is set
r is non empty TopSpace-like TopStruct
the carrier of r is non empty set
[: the carrier of r, the carrier of (TOP-REAL m):] is Relation-like set
bool [: the carrier of r, the carrier of (TOP-REAL m):] is set
g is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
p is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
g <##> p is Relation-like the carrier of r /\ the carrier of r -defined R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m))) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):]
the carrier of r /\ the carrier of r is set
DOMS the carrier of (TOP-REAL m) is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL m) : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL m) : verum } is set
(DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)) is set
R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m))) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):] is Relation-like set
bool [:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):] is set
(m) .: (g,p) is Relation-like Function-like set
<:g,p:> is Relation-like Function-like set
<:g,p:> (#) (m) is Relation-like the carrier of (TOP-REAL m) -valued Function-like Function-yielding V156() complex-functions-valued set
dom (g <##> p) is Element of bool ( the carrier of r /\ the carrier of r)
bool ( the carrier of r /\ the carrier of r) is set
dom g is non empty Element of bool the carrier of r
bool the carrier of r is set
dom p is non empty Element of bool the carrier of r
(dom g) /\ (dom p) is Element of bool the carrier of r
dom (m) is non empty Element of bool the carrier of [:(TOP-REAL m),(TOP-REAL m):]
bool the carrier of [:(TOP-REAL m),(TOP-REAL m):] is set
[: the carrier of (TOP-REAL m), the carrier of (TOP-REAL m):] is Relation-like set
rng g is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL m)
bool the carrier of (TOP-REAL m) is set
rng p is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL m)
[:(rng g),(rng p):] is Relation-like Element of bool the carrier of [:(TOP-REAL m),(TOP-REAL m):]
dom ((m) .: (g,p)) is set
r is set
g . r is Relation-like Function-like complex-valued ext-real-valued real-valued set
p . r is Relation-like Function-like complex-valued ext-real-valued real-valued set
(g <##> p) . r is Relation-like Function-like complex-valued ext-real-valued real-valued set
(g . r) (#) (p . r) is Relation-like Function-like complex-valued ext-real-valued real-valued set
(m) . ((g . r),(p . r)) is set
[(g . r),(p . r)] is set
{(g . r),(p . r)} is functional non empty finite complex-functions-membered ext-real-functions-membered real-functions-membered set
{(g . r)} is functional non empty trivial finite 1 -element complex-functions-membered ext-real-functions-membered real-functions-membered set
{{(g . r),(p . r)},{(g . r)}} is non empty finite V49() set
(m) . [(g . r),(p . r)] is Relation-like Function-like complex-valued ext-real-valued real-valued set
((m) .: (g,p)) . r is set
m is ext-real non negative ordinal natural complex real finite cardinal set
r is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (TOP-REAL m), the carrier of R^1:] is Relation-like complex-valued ext-real-valued real-valued set
bool [: the carrier of (TOP-REAL m), the carrier of R^1:] is set
T is Relation-like the carrier of (TOP-REAL m) -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of (TOP-REAL m), the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of (TOP-REAL m), the carrier of R^1:]
f is Relation-like the carrier of (TOP-REAL m) -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of (TOP-REAL m), the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of (TOP-REAL m), the carrier of R^1:]
g is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
T . g is ext-real complex real set
f . g is ext-real complex real set
T . g is ext-real complex real Element of the carrier of R^1
g /. r is ext-real complex real Element of REAL
f . g is ext-real complex real Element of the carrier of R^1
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
r is ext-real non negative ordinal natural complex real finite cardinal set
(m,r) is Relation-like the carrier of (TOP-REAL m) -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of (TOP-REAL m), the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of (TOP-REAL m), the carrier of R^1:]
[: the carrier of (TOP-REAL m), the carrier of R^1:] is Relation-like complex-valued ext-real-valued real-valued set
bool [: the carrier of (TOP-REAL m), the carrier of R^1:] is set
T is ext-real complex real set
f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
dom f is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
Ball (f,T) is functional complex-functions-membered ext-real-functions-membered real-functions-membered open V269( TOP-REAL m) Element of bool the carrier of (TOP-REAL m)
bool the carrier of (TOP-REAL m) is set
(m,r) .: (Ball (f,T)) is complex-membered ext-real-membered real-membered Element of bool the carrier of R^1
bool the carrier of R^1 is set
f /. r is ext-real complex real Element of REAL
(f /. r) - T is ext-real complex real set
- T is ext-real complex real set
(f /. r) + (- T) is ext-real complex real set
(f /. r) + T is ext-real complex real set
].((f /. r) - T),((f /. r) + T).[ is complex-membered ext-real-membered real-membered open Element of bool REAL
Seg m is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
g is set
p is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(m,r) . p is ext-real complex real Element of the carrier of R^1
p /. r is ext-real complex real Element of REAL
p - f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) f is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K260((- 1)) is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K857() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined Function-like constant non empty total complex-valued ext-real-valued real-valued V315() V316() set
{(- 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(dom (id REAL)),{(- 1)}:] is Relation-like complex-valued ext-real-valued real-valued set
<:((dom (id REAL)) --> (- 1)),(id REAL):> is Relation-like Function-like set
<:((dom (id REAL)) --> (- 1)),(id REAL):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
f (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
K853() is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
f (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
p + (- f) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
p + (- f) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K855() .: (p,(- f)) is Relation-like Function-like set
p - f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
- f is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
p + (- f) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K855() .: (p,(- f)) is Relation-like Function-like set
K856() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K342(REAL,K855(),(id REAL),K853()) is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K856() .: (p,f) is Relation-like Function-like set
<:p,f:> is Relation-like Function-like set
<:p,f:> (#) K856() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
|.(p - f).| is ext-real non negative complex real Element of REAL
sqr (p - f) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(p - f) (#) (p - f) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K857() .: ((p - f),(p - f)) is Relation-like Function-like set
<:(p - f),(p - f):> is Relation-like Function-like set
<:(p - f),(p - f):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(p - f) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum (sqr (p - f)) is ext-real complex real Element of REAL
sqrt (Sum (sqr (p - f))) is ext-real complex real Element of REAL
(m,(p - f)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
Sum (m,(p - f)) is ext-real complex real set
sqrt (Sum (m,(p - f))) is ext-real complex real set
(sqrt (Sum (m,(p - f)))) ^2 is ext-real non negative complex real set
(sqrt (Sum (m,(p - f)))) * (sqrt (Sum (m,(p - f)))) is ext-real non negative complex real set
T ^2 is ext-real non negative complex real set
T * T is ext-real non negative complex real set
dom p is finite m -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
(p /. r) - (f /. r) is ext-real complex real set
- (f /. r) is ext-real complex real set
(p /. r) + (- (f /. r)) is ext-real complex real set
((p /. r) - (f /. r)) ^2 is ext-real non negative complex real set
((p /. r) - (f /. r)) * ((p /. r) - (f /. r)) is ext-real non negative complex real set
((p /. r) - (f /. r)) + (f /. r) is ext-real complex real set
(- T) + (f /. r) is ext-real complex real set
T + (f /. r) is ext-real complex real set
g is set
p is ext-real complex real Element of REAL
(m,f,r,p) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() FinSequence of REAL
dom q is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
(m,f,r) is ext-real complex real Element of REAL
p - (f /. r) is ext-real complex real set
- (f /. r) is ext-real complex real set
p + (- (f /. r)) is ext-real complex real set
(m,f,r,p) - f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) f is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K260((- 1)) is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K857() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined Function-like constant non empty total complex-valued ext-real-valued real-valued V315() V316() set
{(- 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(dom (id REAL)),{(- 1)}:] is Relation-like complex-valued ext-real-valued real-valued set
<:((dom (id REAL)) --> (- 1)),(id REAL):> is Relation-like Function-like set
<:((dom (id REAL)) --> (- 1)),(id REAL):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
f (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
K853() is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
f (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
(m,f,r,p) + (- f) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(m,f,r,p) + (- f) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K855() .: ((m,f,r,p),(- f)) is Relation-like Function-like set
(m,f,r,p) - f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
- f is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
(m,f,r,p) + (- f) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K855() .: ((m,f,r,p),(- f)) is Relation-like Function-like set
K856() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K342(REAL,K855(),(id REAL),K853()) is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K856() .: ((m,f,r,p),f) is Relation-like Function-like set
<:(m,f,r,p),f:> is Relation-like Function-like set
<:(m,f,r,p),f:> (#) K856() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
0* m is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of REAL m
REAL m is functional non empty FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V268() Element of REAL * : len b₁ = m } is set
m |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of m -tuples_on REAL
(Seg m) --> 0 is Relation-like Seg m -defined Seg m -defined RAT -valued INT -valued {0} -valued Function-like constant total total V18( Seg m,{0}) complex-valued ext-real-valued real-valued natural-valued finite FinSequence-like FinSubsequence-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued V268() V316() V317() Element of bool [:(Seg m),{0}:]
{0} is functional non empty trivial finite V49() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered complex-functions-membered ext-real-functions-membered real-functions-membered set
[:(Seg m),{0}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
bool [:(Seg m),{0}:] is finite V49() set
p - (m,f,r) is ext-real complex real set
- (m,f,r) is ext-real complex real set
p + (- (m,f,r)) is ext-real complex real set
(0* m) +* (r,(p - (m,f,r))) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
|.((m,f,r,p) - f).| is ext-real non negative complex real Element of REAL
sqr ((m,f,r,p) - f) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
((m,f,r,p) - f) (#) ((m,f,r,p) - f) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K857() .: (((m,f,r,p) - f),((m,f,r,p) - f)) is Relation-like Function-like set
<:((m,f,r,p) - f),((m,f,r,p) - f):> is Relation-like Function-like set
<:((m,f,r,p) - f),((m,f,r,p) - f):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
((m,f,r,p) - f) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum (sqr ((m,f,r,p) - f)) is ext-real complex real Element of REAL
sqrt (Sum (sqr ((m,f,r,p) - f))) is ext-real complex real Element of REAL
abs (p - (m,f,r)) is ext-real complex real Element of REAL
(m,r) . (m,f,r,p) is ext-real complex real Element of the carrier of R^1
(m,f,r,p) /. r is ext-real complex real Element of REAL
q . r is ext-real complex real set
m is non empty TopSpace-like TopStruct
the carrier of m is non empty set
[: the carrier of m, the carrier of R^1:] is Relation-like complex-valued ext-real-valued real-valued set
bool [: the carrier of m, the carrier of R^1:] is set
r is non empty ext-real positive non negative ordinal natural complex real finite cardinal set
TOP-REAL r is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
(r,r) is Relation-like the carrier of (TOP-REAL r) -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of (TOP-REAL r), the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of (TOP-REAL r), the carrier of R^1:]
[: the carrier of (TOP-REAL r), the carrier of R^1:] is Relation-like complex-valued ext-real-valued real-valued set
bool [: the carrier of (TOP-REAL r), the carrier of R^1:] is set
T is Relation-like the carrier of m -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of m, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m, the carrier of R^1:]
(r, the carrier of m, the carrier of R^1,T) is Relation-like the carrier of m -defined the carrier of (TOP-REAL r) -valued Function-like non empty total V18( the carrier of m, the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of m, the carrier of (TOP-REAL r):]
[: the carrier of m, the carrier of (TOP-REAL r):] is Relation-like set
bool [: the carrier of m, the carrier of (TOP-REAL r):] is set
(r,r) * (r, the carrier of m, the carrier of R^1,T) is Relation-like the carrier of m -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of m, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m, the carrier of R^1:]
f is Element of the carrier of m
T . f is ext-real complex real set
((r,r) * (r, the carrier of m, the carrier of R^1,T)) . f is ext-real complex real set
T . f is ext-real complex real Element of the carrier of R^1
r |-> (T . f) is Relation-like NAT -defined the carrier of R^1 -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of r -tuples_on the carrier of R^1
r -tuples_on the carrier of R^1 is functional non empty FinSequence-membered FinSequenceSet of the carrier of R^1
the carrier of R^1 * is functional non empty FinSequence-membered FinSequenceSet of the carrier of R^1
{ b₁ where b₁ is Relation-like NAT -defined the carrier of R^1 -valued Function-like finite FinSequence-like FinSubsequence-like V268() Element of the carrier of R^1 * : len b₁ = r } is set
Seg r is non empty finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= r ) } is set
(Seg r) --> (T . f) is Relation-like Seg r -defined Seg r -defined the carrier of R^1 -valued {(T . f)} -valued Function-like constant non empty total total V18( Seg r,{(T . f)}) complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() Element of bool [:(Seg r),{(T . f)}:]
{(T . f)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(Seg r),{(T . f)}:] is Relation-like complex-valued ext-real-valued real-valued finite set
bool [:(Seg r),{(T . f)}:] is finite V49() set
dom (r |-> (T . f)) is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
((r,r) * (r, the carrier of m, the carrier of R^1,T)) . f is ext-real complex real Element of the carrier of R^1
(r, the carrier of m, the carrier of R^1,T) . f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
(r,r) . ((r, the carrier of m, the carrier of R^1,T) . f) is ext-real complex real Element of the carrier of R^1
((r, the carrier of m, the carrier of R^1,T) . f) /. r is ext-real complex real Element of REAL
p is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() FinSequence of REAL
p /. r is ext-real complex real Element of REAL
p . r is ext-real complex real set
m is non empty TopSpace-like TopStruct
the carrier of m is non empty set
[: the carrier of m, the carrier of R^1:] is Relation-like complex-valued ext-real-valued real-valued set
bool [: the carrier of m, the carrier of R^1:] is set
R^1 1 is ext-real complex real Element of the carrier of K605()
m --> (R^1 1) is Relation-like the carrier of m -defined the carrier of K605() -valued Function-like non empty total V18( the carrier of m, the carrier of K605()) complex-valued ext-real-valued real-valued continuous Element of bool [: the carrier of m, the carrier of K605():]
[: the carrier of m, the carrier of K605():] is Relation-like complex-valued ext-real-valued real-valued set
bool [: the carrier of m, the carrier of K605():] is set
the carrier of m --> (R^1 1) is Relation-like the carrier of m -defined the carrier of K605() -valued Function-like constant non empty total V18( the carrier of m, the carrier of K605()) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m, the carrier of K605():]
{(R^1 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[: the carrier of m,{(R^1 1)}:] is Relation-like complex-valued ext-real-valued real-valued set
m is ext-real non negative ordinal natural complex real finite cardinal set
r is ext-real non negative ordinal natural complex real finite cardinal set
Seg r is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= r ) } is set
TOP-REAL r is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
(r,m) is Relation-like the carrier of (TOP-REAL r) -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of (TOP-REAL r), the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of (TOP-REAL r), the carrier of R^1:]
the carrier of (TOP-REAL r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (TOP-REAL r), the carrier of R^1:] is Relation-like complex-valued ext-real-valued real-valued set
bool [: the carrier of (TOP-REAL r), the carrier of R^1:] is set
T is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
(r,m) . T is ext-real complex real Element of the carrier of R^1
T /. m is ext-real complex real Element of REAL
m is ext-real non negative ordinal natural complex real finite cardinal set
r is ext-real non negative ordinal natural complex real finite cardinal set
Seg r is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= r ) } is set
TOP-REAL r is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
(r,m) is Relation-like the carrier of (TOP-REAL r) -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of (TOP-REAL r), the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of (TOP-REAL r), the carrier of R^1:]
the carrier of (TOP-REAL r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (TOP-REAL r), the carrier of R^1:] is Relation-like complex-valued ext-real-valued real-valued set
bool [: the carrier of (TOP-REAL r), the carrier of R^1:] is set
f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
(r,m) . f is ext-real complex real Element of the carrier of R^1
g is non empty ext-real positive non negative complex real set
Ball (f,g) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered open V269( TOP-REAL r) Element of bool the carrier of (TOP-REAL r)
bool the carrier of (TOP-REAL r) is set
(r,m) .: (Ball (f,g)) is complex-membered ext-real-membered real-membered Element of bool the carrier of R^1
bool the carrier of R^1 is set
((r,m) . f) - g is ext-real complex real set
- g is non empty ext-real non positive negative complex real set
((r,m) . f) + (- g) is ext-real complex real set
((r,m) . f) + g is ext-real complex real set
].(((r,m) . f) - g),(((r,m) . f) + g).[ is complex-membered ext-real-membered real-membered open Element of bool REAL
dom f is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
f /. m is ext-real complex real Element of REAL
r is non empty TopSpace-like TopStruct
the carrier of r is non empty set
[: the carrier of r, the carrier of R^1:] is Relation-like complex-valued ext-real-valued real-valued set
bool [: the carrier of r, the carrier of R^1:] is set
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
T is Relation-like the carrier of r -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of r, the carrier of R^1) complex-valued ext-real-valued real-valued continuous Element of bool [: the carrier of r, the carrier of R^1:]
(m, the carrier of r, the carrier of R^1,T) is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of r, the carrier of (TOP-REAL m):] is Relation-like set
bool [: the carrier of r, the carrier of (TOP-REAL m):] is set
f is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
(f, the carrier of r, the carrier of R^1,T) is Relation-like the carrier of r -defined the carrier of (TOP-REAL f) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL f)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL f):]
TOP-REAL f is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL f) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of r, the carrier of (TOP-REAL f):] is Relation-like set
bool [: the carrier of r, the carrier of (TOP-REAL f):] is set
(0, the carrier of r, the carrier of R^1,T) is Relation-like the carrier of r -defined the carrier of (TOP-REAL 0) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL 0)) complex-valued ext-real-valued real-valued Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL 0):]
[: the carrier of r, the carrier of (TOP-REAL 0):] is Relation-like complex-valued ext-real-valued real-valued set
bool [: the carrier of r, the carrier of (TOP-REAL 0):] is set
p is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite f -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL f)
r --> p is Relation-like the carrier of r -defined the carrier of (TOP-REAL f) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL f)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued continuous Element of bool [: the carrier of r, the carrier of (TOP-REAL f):]
the carrier of r --> p is Relation-like the carrier of r -defined the carrier of (TOP-REAL f) -valued Function-like constant non empty total V18( the carrier of r, the carrier of (TOP-REAL f)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL f):]
{p} is functional non empty trivial finite V49() 1 -element complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of r,{p}:] is Relation-like set
bool the carrier of (TOP-REAL f) is set
bool the carrier of r is set
p is Element of the carrier of r
(f, the carrier of r, the carrier of R^1,T) . p is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite f -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL f)
r is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL f)
Int r is functional complex-functions-membered ext-real-functions-membered real-functions-membered open Element of bool the carrier of (TOP-REAL f)
Euclid f is non empty V138() V139() strict Reflexive discerning V193() triangle () () MetrStruct
REAL f is functional non empty FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
f -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V268() Element of REAL * : len b₁ = f } is set
Pitag_dist f is Relation-like [:(REAL f),(REAL f):] -defined REAL -valued Function-like total V18([:(REAL f),(REAL f):], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:(REAL f),(REAL f):],REAL:]
[:(REAL f),(REAL f):] is Relation-like set
[:[:(REAL f),(REAL f):],REAL:] is Relation-like complex-valued ext-real-valued real-valued set
bool [:[:(REAL f),(REAL f):],REAL:] is set
MetrStruct(# (REAL f),(Pitag_dist f) #) is strict MetrStruct
the carrier of (Euclid f) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite f -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (Euclid f)
W is ext-real complex real set
Ball (q,W) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Euclid f)
bool the carrier of (Euclid f) is set
T . p is ext-real complex real Element of the carrier of R^1
f |-> (T . p) is Relation-like NAT -defined the carrier of R^1 -valued Function-like complex-valued ext-real-valued real-valued finite f -element FinSequence-like FinSubsequence-like V268() Element of f -tuples_on the carrier of R^1
f -tuples_on the carrier of R^1 is functional non empty FinSequence-membered FinSequenceSet of the carrier of R^1
the carrier of R^1 * is functional non empty FinSequence-membered FinSequenceSet of the carrier of R^1
{ b₁ where b₁ is Relation-like NAT -defined the carrier of R^1 -valued Function-like finite FinSequence-like FinSubsequence-like V268() Element of the carrier of R^1 * : len b₁ = f } is set
Seg f is finite f -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= f ) } is set
(Seg f) --> (T . p) is Relation-like Seg f -defined Seg f -defined the carrier of R^1 -valued {(T . p)} -valued Function-like constant total total V18( Seg f,{(T . p)}) complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() Element of bool [:(Seg f),{(T . p)}:]
{(T . p)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(Seg f),{(T . p)}:] is Relation-like complex-valued ext-real-valued real-valued finite set
bool [:(Seg f),{(T . p)}:] is finite V49() set
y is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() FinSequence of REAL
dom (f |-> (T . p)) is finite f -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
((f, the carrier of r, the carrier of R^1,T) . p) /. f is ext-real complex real Element of REAL
W1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() FinSequence of REAL
W1 /. f is ext-real complex real Element of REAL
W1 . f is ext-real complex real set
sqrt f is ext-real non negative complex real set
W / (sqrt f) is ext-real complex real Element of COMPLEX
(sqrt f) " is ext-real non negative complex real set
W * ((sqrt f) ") is ext-real complex real set
(((f, the carrier of r, the carrier of R^1,T) . p) /. f) - (W / (sqrt f)) is ext-real complex real set
- (W / (sqrt f)) is ext-real complex real set
(((f, the carrier of r, the carrier of R^1,T) . p) /. f) + (- (W / (sqrt f))) is ext-real complex real set
(((f, the carrier of r, the carrier of R^1,T) . p) /. f) + (W / (sqrt f)) is ext-real complex real set
].((((f, the carrier of r, the carrier of R^1,T) . p) /. f) - (W / (sqrt f))),((((f, the carrier of r, the carrier of R^1,T) . p) /. f) + (W / (sqrt f))).[ is complex-membered ext-real-membered real-membered open Element of bool REAL
R^1 ].((((f, the carrier of r, the carrier of R^1,T) . p) /. f) - (W / (sqrt f))),((((f, the carrier of r, the carrier of R^1,T) . p) /. f) + (W / (sqrt f))).[ is complex-membered ext-real-membered real-membered open Element of bool the carrier of K605()
G2 is Element of bool the carrier of r
T .: G2 is complex-membered ext-real-membered real-membered Element of bool the carrier of R^1
bool the carrier of R^1 is set
(f, the carrier of r, the carrier of R^1,T) .: G2 is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL f)
W2 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite f -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL f)
W is Element of the carrier of r
(f, the carrier of r, the carrier of R^1,T) . W is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite f -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL f)
x is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite f -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (Euclid f)
z is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite f -element FinSequence-like FinSubsequence-like V268() Element of REAL f
T . W is ext-real complex real Element of the carrier of R^1
f |-> (T . W) is Relation-like NAT -defined the carrier of R^1 -valued Function-like complex-valued ext-real-valued real-valued finite f -element FinSequence-like FinSubsequence-like V268() Element of f -tuples_on the carrier of R^1
(Seg f) --> (T . W) is Relation-like Seg f -defined Seg f -defined the carrier of R^1 -valued {(T . W)} -valued Function-like constant total total V18( Seg f,{(T . W)}) complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() Element of bool [:(Seg f),{(T . W)}:]
{(T . W)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(Seg f),{(T . W)}:] is Relation-like complex-valued ext-real-valued real-valued finite set
bool [:(Seg f),{(T . W)}:] is finite V49() set
(Pitag_dist f) . (x,q) is set
[x,q] is set
{x,q} is functional non empty finite V49() complex-functions-membered ext-real-functions-membered real-functions-membered set
{x} is functional non empty trivial finite V49() 1 -element complex-functions-membered ext-real-functions-membered real-functions-membered set
{{x,q},{x}} is non empty finite V49() set
(Pitag_dist f) . [x,q] is ext-real complex real set
pz is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite f -element FinSequence-like FinSubsequence-like V268() Element of REAL f
z - pz is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite f -element FinSequence-like FinSubsequence-like V268() Element of REAL f
- pz is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite f -element FinSequence-like FinSubsequence-like V268() set
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) pz is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite f -element FinSequence-like FinSubsequence-like V268() set
K260((- 1)) is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K857() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined Function-like constant non empty total complex-valued ext-real-valued real-valued V315() V316() set
{(- 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(dom (id REAL)),{(- 1)}:] is Relation-like complex-valued ext-real-valued real-valued set
<:((dom (id REAL)) --> (- 1)),(id REAL):> is Relation-like Function-like set
<:((dom (id REAL)) --> (- 1)),(id REAL):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
pz (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
K853() is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
pz (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
z + (- pz) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite f -element FinSequence-like FinSubsequence-like V268() set
K855() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K855() .: (z,(- pz)) is Relation-like Function-like set
<:z,(- pz):> is Relation-like Function-like set
<:z,(- pz):> (#) K855() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
K856() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K342(REAL,K855(),(id REAL),K853()) is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K856() .: (z,pz) is Relation-like Function-like set
<:z,pz:> is Relation-like Function-like set
<:z,pz:> (#) K856() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
|.(z - pz).| is ext-real non negative complex real Element of REAL
sqr (z - pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite f -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(z - pz) (#) (z - pz) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite f -element FinSequence-like FinSubsequence-like V268() V317() set
K857() .: ((z - pz),(z - pz)) is Relation-like Function-like set
<:(z - pz),(z - pz):> is Relation-like Function-like set
<:(z - pz),(z - pz):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(z - pz) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum (sqr (z - pz)) is ext-real complex real Element of REAL
sqrt (Sum (sqr (z - pz))) is ext-real complex real Element of REAL
(T . W) - (T . p) is ext-real complex real set
- (T . p) is ext-real complex real set
(T . W) + (- (T . p)) is ext-real complex real set
abs ((T . W) - (T . p)) is ext-real complex real Element of REAL
(sqrt f) * (abs ((T . W) - (T . p))) is ext-real complex real set
dist (x,q) is ext-real complex real Element of REAL
the distance of (Euclid f) is Relation-like [: the carrier of (Euclid f), the carrier of (Euclid f):] -defined REAL -valued Function-like total V18([: the carrier of (Euclid f), the carrier of (Euclid f):], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[: the carrier of (Euclid f), the carrier of (Euclid f):],REAL:]
[: the carrier of (Euclid f), the carrier of (Euclid f):] is Relation-like set
[:[: the carrier of (Euclid f), the carrier of (Euclid f):],REAL:] is Relation-like complex-valued ext-real-valued real-valued set
bool [:[: the carrier of (Euclid f), the carrier of (Euclid f):],REAL:] is set
the distance of (Euclid f) . (x,q) is ext-real complex real Element of REAL
the distance of (Euclid f) . [x,q] is ext-real complex real set
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
[:(TOP-REAL m),(TOP-REAL m):] is non empty strict TopSpace-like TopStruct
(m) is Relation-like the carrier of [:(TOP-REAL m),(TOP-REAL m):] -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of [:(TOP-REAL m),(TOP-REAL m):], the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of [:(TOP-REAL m),(TOP-REAL m):], the carrier of (TOP-REAL m):]
the carrier of [:(TOP-REAL m),(TOP-REAL m):] is non empty set
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of [:(TOP-REAL m),(TOP-REAL m):], the carrier of (TOP-REAL m):] is Relation-like set
bool [: the carrier of [:(TOP-REAL m),(TOP-REAL m):], the carrier of (TOP-REAL m):] is set
r is non empty ext-real positive non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT
TOP-REAL r is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
(r) is Relation-like the carrier of [:(TOP-REAL r),(TOP-REAL r):] -defined the carrier of (TOP-REAL r) -valued Function-like non empty total V18( the carrier of [:(TOP-REAL r),(TOP-REAL r):], the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of [:(TOP-REAL r),(TOP-REAL r):], the carrier of (TOP-REAL r):]
[:(TOP-REAL r),(TOP-REAL r):] is non empty strict TopSpace-like TopStruct
the carrier of [:(TOP-REAL r),(TOP-REAL r):] is non empty set
the carrier of (TOP-REAL r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of [:(TOP-REAL r),(TOP-REAL r):], the carrier of (TOP-REAL r):] is Relation-like set
bool [: the carrier of [:(TOP-REAL r),(TOP-REAL r):], the carrier of (TOP-REAL r):] is set
Seg r is non empty finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
{ b₁ where b₁ is ext-real non negative ordinal natural complex real V43() V44() finite cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of NAT : ( 1 <= b₁ & b₁ <= r ) } is set
(Seg r) --> R^1 is Relation-like Seg r -defined Seg r -defined {R^1} -valued Function-like constant non empty total total V18( Seg r,{R^1}) finite FinSequence-like FinSubsequence-like V268() V291() V306() V309() Element of bool [:(Seg r),{R^1}:]
{R^1} is non empty trivial finite 1 -element set
[:(Seg r),{R^1}:] is Relation-like finite set
bool [:(Seg r),{R^1}:] is finite V49() set
Euclid r is non empty V138() V139() strict Reflexive discerning V193() triangle () () MetrStruct
REAL r is functional non empty FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
r -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V268() Element of REAL * : len b₁ = r } is set
Pitag_dist r is Relation-like [:(REAL r),(REAL r):] -defined REAL -valued Function-like total V18([:(REAL r),(REAL r):], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:(REAL r),(REAL r):],REAL:]
[:(REAL r),(REAL r):] is Relation-like set
[:[:(REAL r),(REAL r):],REAL:] is Relation-like complex-valued ext-real-valued real-valued set
bool [:[:(REAL r),(REAL r):],REAL:] is set
MetrStruct(# (REAL r),(Pitag_dist r) #) is strict MetrStruct
TopSpaceMetr (Euclid r) is TopStruct
the carrier of (Euclid r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
Family_open_set (Euclid r) is Element of bool (bool the carrier of (Euclid r))
bool the carrier of (Euclid r) is set
bool (bool the carrier of (Euclid r)) is set
TopStruct(# the carrier of (Euclid r),(Family_open_set (Euclid r)) #) is non empty strict TopStruct
product ((Seg r) --> R^1) is non empty V138() strict TopSpace-like TopStruct
the topology of (TOP-REAL r) is non empty Element of bool (bool the carrier of (TOP-REAL r))
bool the carrier of (TOP-REAL r) is set
bool (bool the carrier of (TOP-REAL r)) is set
TopStruct(# the carrier of (TOP-REAL r), the topology of (TOP-REAL r) #) is non empty strict TopSpace-like TopStruct
the carrier of (product ((Seg r) --> R^1)) is non empty set
[: the carrier of [:(TOP-REAL r),(TOP-REAL r):], the carrier of (product ((Seg r) --> R^1)):] is Relation-like set
bool [: the carrier of [:(TOP-REAL r),(TOP-REAL r):], the carrier of (product ((Seg r) --> R^1)):] is set
q is ext-real non negative ordinal natural complex real V43() V44() finite cardinal Element of Seg r
proj (((Seg r) --> R^1),q) is Relation-like the carrier of (product ((Seg r) --> R^1)) -defined the carrier of (product ((Seg r) --> R^1)) -defined the carrier of (((Seg r) --> R^1) . q) -valued the carrier of (((Seg r) --> R^1) . q) -valued Function-like non empty total total V18( the carrier of (product ((Seg r) --> R^1)), the carrier of (((Seg r) --> R^1) . q)) V18( the carrier of (product ((Seg r) --> R^1)), the carrier of (((Seg r) --> R^1) . q)) complex-valued ext-real-valued real-valued Element of bool [: the carrier of (product ((Seg r) --> R^1)), the carrier of (((Seg r) --> R^1) . q):]
((Seg r) --> R^1) . q is non empty TopStruct
the carrier of (((Seg r) --> R^1) . q) is non empty set
[: the carrier of (product ((Seg r) --> R^1)), the carrier of (((Seg r) --> R^1) . q):] is Relation-like set
bool [: the carrier of (product ((Seg r) --> R^1)), the carrier of (((Seg r) --> R^1) . q):] is set
[: the carrier of [:(TOP-REAL r),(TOP-REAL r):], the carrier of R^1:] is Relation-like complex-valued ext-real-valued real-valued set
bool [: the carrier of [:(TOP-REAL r),(TOP-REAL r):], the carrier of R^1:] is set
r is Relation-like the carrier of [:(TOP-REAL r),(TOP-REAL r):] -defined the carrier of (product ((Seg r) --> R^1)) -valued Function-like non empty total V18( the carrier of [:(TOP-REAL r),(TOP-REAL r):], the carrier of (product ((Seg r) --> R^1))) Element of bool [: the carrier of [:(TOP-REAL r),(TOP-REAL r):], the carrier of (product ((Seg r) --> R^1)):]
(proj (((Seg r) --> R^1),q)) * r is Relation-like the carrier of [:(TOP-REAL r),(TOP-REAL r):] -defined the carrier of (((Seg r) --> R^1) . q) -valued Function-like non empty total V18( the carrier of [:(TOP-REAL r),(TOP-REAL r):], the carrier of (((Seg r) --> R^1) . q)) complex-valued ext-real-valued real-valued Element of bool [: the carrier of [:(TOP-REAL r),(TOP-REAL r):], the carrier of (((Seg r) --> R^1) . q):]
[: the carrier of [:(TOP-REAL r),(TOP-REAL r):], the carrier of (((Seg r) --> R^1) . q):] is Relation-like set
bool [: the carrier of [:(TOP-REAL r),(TOP-REAL r):], the carrier of (((Seg r) --> R^1) . q):] is set
[: the carrier of (TOP-REAL r), the carrier of (TOP-REAL r):] is Relation-like set
W1 is Relation-like the carrier of [:(TOP-REAL r),(TOP-REAL r):] -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of [:(TOP-REAL r),(TOP-REAL r):], the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of [:(TOP-REAL r),(TOP-REAL r):], the carrier of R^1:]
y is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
x is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
((TOP-REAL r),(TOP-REAL r),R^1,W1,y,x) is ext-real complex real Element of the carrier of R^1
[y,x] is set
{y,x} is functional non empty finite V49() complex-functions-membered ext-real-functions-membered real-functions-membered set
{y} is functional non empty trivial finite V49() 1 -element complex-functions-membered ext-real-functions-membered real-functions-membered set
{{y,x},{y}} is non empty finite V49() set
W1 . [y,x] is ext-real complex real set
(r,y,q) is ext-real complex real Element of REAL
(r,x,q) is ext-real complex real Element of REAL
(r,y,q) * (r,x,q) is ext-real complex real set
((TOP-REAL r),(TOP-REAL r),(product ((Seg r) --> R^1)),r,y,x) is Relation-like Function-like Element of the carrier of (product ((Seg r) --> R^1))
r . [y,x] is set
(proj (((Seg r) --> R^1),q)) . ((TOP-REAL r),(TOP-REAL r),(product ((Seg r) --> R^1)),r,y,x) is ext-real complex real Element of the carrier of (((Seg r) --> R^1) . q)
(r,y,x) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
K857() .: (y,x) is Relation-like Function-like set
<:y,x:> is Relation-like Function-like set
<:y,x:> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(proj (((Seg r) --> R^1),q)) . (r,y,x) is ext-real complex real set
(r,(r,y,x),q) is ext-real complex real Element of REAL
[:[: the carrier of (TOP-REAL r), the carrier of (TOP-REAL r):],REAL:] is Relation-like complex-valued ext-real-valued real-valued set
bool [:[: the carrier of (TOP-REAL r), the carrier of (TOP-REAL r):],REAL:] is set
y is Relation-like [: the carrier of (TOP-REAL r), the carrier of (TOP-REAL r):] -defined REAL -valued Function-like total V18([: the carrier of (TOP-REAL r), the carrier of (TOP-REAL r):], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[: the carrier of (TOP-REAL r), the carrier of (TOP-REAL r):],REAL:]
r02 is Relation-like [: the carrier of (TOP-REAL r), the carrier of (TOP-REAL r):] -defined REAL -valued Function-like total V18([: the carrier of (TOP-REAL r), the carrier of (TOP-REAL r):], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[: the carrier of (TOP-REAL r), the carrier of (TOP-REAL r):],REAL:]
x is Relation-like the carrier of [:(TOP-REAL r),(TOP-REAL r):] -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of [:(TOP-REAL r),(TOP-REAL r):], the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of [:(TOP-REAL r),(TOP-REAL r):], the carrier of R^1:]
bool the carrier of [:(TOP-REAL r),(TOP-REAL r):] is set
G2 is Relation-like the carrier of [:(TOP-REAL r),(TOP-REAL r):] -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of [:(TOP-REAL r),(TOP-REAL r):], the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of [:(TOP-REAL r),(TOP-REAL r):], the carrier of R^1:]
W is Element of the carrier of [:(TOP-REAL r),(TOP-REAL r):]
((TOP-REAL r),(TOP-REAL r),R^1,G2,W) is ext-real complex real Element of the carrier of R^1
x is non empty ext-real positive non negative complex real set
((TOP-REAL r),(TOP-REAL r),R^1,G2,W) - x is ext-real complex real set
- x is non empty ext-real non positive negative complex real set
((TOP-REAL r),(TOP-REAL r),R^1,G2,W) + (- x) is ext-real complex real set
((TOP-REAL r),(TOP-REAL r),R^1,G2,W) + x is ext-real complex real set
].(((TOP-REAL r),(TOP-REAL r),R^1,G2,W) - x),(((TOP-REAL r),(TOP-REAL r),R^1,G2,W) + x).[ is complex-membered ext-real-membered real-membered open Element of bool REAL
z is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
pz is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
[z,pz] is Element of the carrier of [:(TOP-REAL r),(TOP-REAL r):]
{z,pz} is functional non empty finite V49() complex-functions-membered ext-real-functions-membered real-functions-membered set
{z} is functional non empty trivial finite V49() 1 -element complex-functions-membered ext-real-functions-membered real-functions-membered set
{{z,pz},{z}} is non empty finite V49() set
Ball (z,x) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered open V215( TOP-REAL r) V269( TOP-REAL r) Element of bool the carrier of (TOP-REAL r)
[#] (TOP-REAL r) is functional non empty non proper complex-functions-membered ext-real-functions-membered real-functions-membered open closed dense non boundary Element of bool the carrier of (TOP-REAL r)
[:(Ball (z,x)),([#] (TOP-REAL r)):] is Relation-like Element of bool the carrier of [:(TOP-REAL r),(TOP-REAL r):]
bb0 is open Element of bool the carrier of [:(TOP-REAL r),(TOP-REAL r):]
G2 .: bb0 is complex-membered ext-real-membered real-membered Element of bool the carrier of R^1
bool the carrier of R^1 is set
y is set
x is Element of the carrier of [:(TOP-REAL r),(TOP-REAL r):]
((TOP-REAL r),(TOP-REAL r),R^1,G2,x) is ext-real complex real Element of the carrier of R^1
x1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
x2 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
[x1,x2] is Element of the carrier of [:(TOP-REAL r),(TOP-REAL r):]
{x1,x2} is functional non empty finite V49() complex-functions-membered ext-real-functions-membered real-functions-membered set
{x1} is functional non empty trivial finite V49() 1 -element complex-functions-membered ext-real-functions-membered real-functions-membered set
{{x1,x2},{x1}} is non empty finite V49() set
((TOP-REAL r),(TOP-REAL r),R^1,G2,x1,x2) is ext-real complex real Element of the carrier of R^1
[x1,x2] is set
G2 . [x1,x2] is ext-real complex real set
(r,x1,q) is ext-real complex real Element of REAL
((TOP-REAL r),(TOP-REAL r),R^1,G2,z,pz) is ext-real complex real Element of the carrier of R^1
[z,pz] is set
G2 . [z,pz] is ext-real complex real set
(r,z,q) is ext-real complex real Element of REAL
x1 - z is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
- z is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
- z is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) z is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
K260((- 1)) is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K857() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined Function-like constant non empty total complex-valued ext-real-valued real-valued V315() V316() set
{(- 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(dom (id REAL)),{(- 1)}:] is Relation-like complex-valued ext-real-valued real-valued set
<:((dom (id REAL)) --> (- 1)),(id REAL):> is Relation-like Function-like set
<:((dom (id REAL)) --> (- 1)),(id REAL):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
z (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
K853() is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
z (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
x1 + (- z) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
x1 + (- z) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K855() .: (x1,(- z)) is Relation-like Function-like set
x1 - z is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
- z is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
x1 + (- z) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
K855() .: (x1,(- z)) is Relation-like Function-like set
K856() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K342(REAL,K855(),(id REAL),K853()) is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K856() .: (x1,z) is Relation-like Function-like set
<:x1,z:> is Relation-like Function-like set
<:x1,z:> (#) K856() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
|.(x1 - z).| is ext-real non negative complex real Element of REAL
sqr (x1 - z) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(x1 - z) (#) (x1 - z) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
K857() .: ((x1 - z),(x1 - z)) is Relation-like Function-like set
<:(x1 - z),(x1 - z):> is Relation-like Function-like set
<:(x1 - z),(x1 - z):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(x1 - z) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum (sqr (x1 - z)) is ext-real complex real Element of REAL
sqrt (Sum (sqr (x1 - z))) is ext-real complex real Element of REAL
dom (x1 - z) is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
(r,x1,q) - (r,z,q) is ext-real complex real set
- (r,z,q) is ext-real complex real set
(r,x1,q) + (- (r,z,q)) is ext-real complex real set
(r,(x1 - z),q) is ext-real complex real Element of REAL
abs ((r,x1,q) - (r,z,q)) is ext-real complex real Element of REAL
((TOP-REAL r),(TOP-REAL r),R^1,G2,x) - ((TOP-REAL r),(TOP-REAL r),R^1,G2,W) is ext-real complex real set
- ((TOP-REAL r),(TOP-REAL r),R^1,G2,W) is ext-real complex real set
((TOP-REAL r),(TOP-REAL r),R^1,G2,x) + (- ((TOP-REAL r),(TOP-REAL r),R^1,G2,W)) is ext-real complex real set
abs (((TOP-REAL r),(TOP-REAL r),R^1,G2,x) - ((TOP-REAL r),(TOP-REAL r),R^1,G2,W)) is ext-real complex real Element of REAL
W2 is Relation-like the carrier of [:(TOP-REAL r),(TOP-REAL r):] -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of [:(TOP-REAL r),(TOP-REAL r):], the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of [:(TOP-REAL r),(TOP-REAL r):], the carrier of R^1:]
W is Element of the carrier of [:(TOP-REAL r),(TOP-REAL r):]
((TOP-REAL r),(TOP-REAL r),R^1,W2,W) is ext-real complex real Element of the carrier of R^1
x is non empty ext-real positive non negative complex real set
((TOP-REAL r),(TOP-REAL r),R^1,W2,W) - x is ext-real complex real set
- x is non empty ext-real non positive negative complex real set
((TOP-REAL r),(TOP-REAL r),R^1,W2,W) + (- x) is ext-real complex real set
((TOP-REAL r),(TOP-REAL r),R^1,W2,W) + x is ext-real complex real set
].(((TOP-REAL r),(TOP-REAL r),R^1,W2,W) - x),(((TOP-REAL r),(TOP-REAL r),R^1,W2,W) + x).[ is complex-membered ext-real-membered real-membered open Element of bool REAL
z is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
pz is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
[z,pz] is Element of the carrier of [:(TOP-REAL r),(TOP-REAL r):]
{z,pz} is functional non empty finite V49() complex-functions-membered ext-real-functions-membered real-functions-membered set
{z} is functional non empty trivial finite V49() 1 -element complex-functions-membered ext-real-functions-membered real-functions-membered set
{{z,pz},{z}} is non empty finite V49() set
[#] (TOP-REAL r) is functional non empty non proper complex-functions-membered ext-real-functions-membered real-functions-membered open closed dense non boundary Element of bool the carrier of (TOP-REAL r)
Ball (pz,x) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered open V215( TOP-REAL r) V269( TOP-REAL r) Element of bool the carrier of (TOP-REAL r)
[:([#] (TOP-REAL r)),(Ball (pz,x)):] is Relation-like Element of bool the carrier of [:(TOP-REAL r),(TOP-REAL r):]
bb0 is open Element of bool the carrier of [:(TOP-REAL r),(TOP-REAL r):]
W2 .: bb0 is complex-membered ext-real-membered real-membered Element of bool the carrier of R^1
bool the carrier of R^1 is set
y is set
x is Element of the carrier of [:(TOP-REAL r),(TOP-REAL r):]
((TOP-REAL r),(TOP-REAL r),R^1,W2,x) is ext-real complex real Element of the carrier of R^1
x1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
x2 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
[x1,x2] is Element of the carrier of [:(TOP-REAL r),(TOP-REAL r):]
{x1,x2} is functional non empty finite V49() complex-functions-membered ext-real-functions-membered real-functions-membered set
{x1} is functional non empty trivial finite V49() 1 -element complex-functions-membered ext-real-functions-membered real-functions-membered set
{{x1,x2},{x1}} is non empty finite V49() set
((TOP-REAL r),(TOP-REAL r),R^1,W2,x1,x2) is ext-real complex real Element of the carrier of R^1
[x1,x2] is set
W2 . [x1,x2] is ext-real complex real set
(r,x2,q) is ext-real complex real Element of REAL
((TOP-REAL r),(TOP-REAL r),R^1,W2,z,pz) is ext-real complex real Element of the carrier of R^1
[z,pz] is set
W2 . [z,pz] is ext-real complex real set
(r,pz,q) is ext-real complex real Element of REAL
x2 - pz is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
- pz is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
- pz is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) pz is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
K260((- 1)) is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K857() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined Function-like constant non empty total complex-valued ext-real-valued real-valued V315() V316() set
{(- 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(dom (id REAL)),{(- 1)}:] is Relation-like complex-valued ext-real-valued real-valued set
<:((dom (id REAL)) --> (- 1)),(id REAL):> is Relation-like Function-like set
<:((dom (id REAL)) --> (- 1)),(id REAL):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
pz (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
K853() is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
pz (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
x2 + (- pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
x2 + (- pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K855() .: (x2,(- pz)) is Relation-like Function-like set
x2 - pz is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
- pz is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
x2 + (- pz) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
K855() .: (x2,(- pz)) is Relation-like Function-like set
K856() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K342(REAL,K855(),(id REAL),K853()) is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K856() .: (x2,pz) is Relation-like Function-like set
<:x2,pz:> is Relation-like Function-like set
<:x2,pz:> (#) K856() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
|.(x2 - pz).| is ext-real non negative complex real Element of REAL
sqr (x2 - pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(x2 - pz) (#) (x2 - pz) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
K857() .: ((x2 - pz),(x2 - pz)) is Relation-like Function-like set
<:(x2 - pz),(x2 - pz):> is Relation-like Function-like set
<:(x2 - pz),(x2 - pz):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(x2 - pz) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum (sqr (x2 - pz)) is ext-real complex real Element of REAL
sqrt (Sum (sqr (x2 - pz))) is ext-real complex real Element of REAL
dom (x2 - pz) is finite r -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of bool NAT
(r,x2,q) - (r,pz,q) is ext-real complex real set
- (r,pz,q) is ext-real complex real set
(r,x2,q) + (- (r,pz,q)) is ext-real complex real set
(r,(x2 - pz),q) is ext-real complex real Element of REAL
abs ((r,x2,q) - (r,pz,q)) is ext-real complex real Element of REAL
((TOP-REAL r),(TOP-REAL r),R^1,W2,x) - ((TOP-REAL r),(TOP-REAL r),R^1,W2,W) is ext-real complex real set
- ((TOP-REAL r),(TOP-REAL r),R^1,W2,W) is ext-real complex real set
((TOP-REAL r),(TOP-REAL r),R^1,W2,x) + (- ((TOP-REAL r),(TOP-REAL r),R^1,W2,W)) is ext-real complex real set
abs (((TOP-REAL r),(TOP-REAL r),R^1,W2,x) - ((TOP-REAL r),(TOP-REAL r),R^1,W2,W)) is ext-real complex real Element of REAL
W is Relation-like the carrier of [:(TOP-REAL r),(TOP-REAL r):] -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of [:(TOP-REAL r),(TOP-REAL r):], the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of [:(TOP-REAL r),(TOP-REAL r):], the carrier of R^1:]
x is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
z is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
((TOP-REAL r),(TOP-REAL r),R^1,G2,x,z) is ext-real complex real Element of the carrier of R^1
[x,z] is set
{x,z} is functional non empty finite V49() complex-functions-membered ext-real-functions-membered real-functions-membered set
{x} is functional non empty trivial finite V49() 1 -element complex-functions-membered ext-real-functions-membered real-functions-membered set
{{x,z},{x}} is non empty finite V49() set
G2 . [x,z] is ext-real complex real set
(r,x,q) is ext-real complex real Element of REAL
((TOP-REAL r),(TOP-REAL r),R^1,W2,x,z) is ext-real complex real Element of the carrier of R^1
W2 . [x,z] is ext-real complex real set
(r,z,q) is ext-real complex real Element of REAL
((TOP-REAL r),(TOP-REAL r),R^1,W1,x,z) is ext-real complex real Element of the carrier of R^1
W1 . [x,z] is ext-real complex real set
(r,x,q) * (r,z,q) is ext-real complex real set
[x,z] is Element of the carrier of [:(TOP-REAL r),(TOP-REAL r):]
((TOP-REAL r),(TOP-REAL r),R^1,W,[x,z]) is ext-real complex real Element of the carrier of R^1
((TOP-REAL r),(TOP-REAL r),R^1,W,x,z) is ext-real complex real Element of the carrier of R^1
W . [x,z] is ext-real complex real set
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
r is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL r is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (TOP-REAL m), the carrier of (TOP-REAL r):] is Relation-like set
bool [: the carrier of (TOP-REAL m), the carrier of (TOP-REAL r):] is set
T is Relation-like the carrier of (TOP-REAL m) -defined the carrier of (TOP-REAL r) -valued Function-like non empty total V18( the carrier of (TOP-REAL m), the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL m), the carrier of (TOP-REAL r):]
T (-) is Relation-like Function-like set
f is Relation-like the carrier of (TOP-REAL m) -defined the carrier of (TOP-REAL r) -valued Function-like non empty total V18( the carrier of (TOP-REAL m), the carrier of (TOP-REAL r)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (TOP-REAL m), the carrier of (TOP-REAL r):]
g is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
f . g is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
p is non empty ext-real positive non negative complex real set
Ball ((f . g),p) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered open V269( TOP-REAL r) Element of bool the carrier of (TOP-REAL r)
bool the carrier of (TOP-REAL r) is set
- g is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- g is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) g is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K260((- 1)) is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K857() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined Function-like constant non empty total complex-valued ext-real-valued real-valued V315() V316() set
{(- 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(dom (id REAL)),{(- 1)}:] is Relation-like complex-valued ext-real-valued real-valued set
<:((dom (id REAL)) --> (- 1)),(id REAL):> is Relation-like Function-like set
<:((dom (id REAL)) --> (- 1)),(id REAL):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
g (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
K853() is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
g (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
T . (- g) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
Ball ((T . (- g)),p) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered open V269( TOP-REAL r) Element of bool the carrier of (TOP-REAL r)
r is non empty ext-real positive non negative complex real set
Ball ((- g),r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered open V269( TOP-REAL m) Element of bool the carrier of (TOP-REAL m)
bool the carrier of (TOP-REAL m) is set
T .: (Ball ((- g),r)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL r)
Ball (g,r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered open V269( TOP-REAL m) Element of bool the carrier of (TOP-REAL m)
f .: (Ball (g,r)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL r)
q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
W is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
f . W is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
dom f is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL m)
- W is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- W is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(- 1) (#) W is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
W (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
W (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
T . (- W) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
m is non empty TopSpace-like TopStruct
the carrier of m is non empty set
[: the carrier of m, the carrier of R^1:] is Relation-like complex-valued ext-real-valued real-valued set
bool [: the carrier of m, the carrier of R^1:] is set
r is Relation-like the carrier of m -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of m, the carrier of R^1) complex-valued ext-real-valued real-valued continuous Element of bool [: the carrier of m, the carrier of R^1:]
- r is Relation-like the carrier of m -defined REAL -valued Function-like non empty total V18( the carrier of m, REAL ) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m,REAL:]
[: the carrier of m,REAL:] is Relation-like non trivial complex-valued ext-real-valued real-valued non finite set
bool [: the carrier of m,REAL:] is non trivial non finite set
- 1 is non empty ext-real non positive negative complex real set
(- 1) (#) r is Relation-like the carrier of m -defined Function-like total complex-valued ext-real-valued real-valued set
T is Relation-like the carrier of m -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of m, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m, the carrier of R^1:]
f is Relation-like the carrier of m -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of m, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m, the carrier of R^1:]
g is Element of the carrier of m
T . g is ext-real complex real set
f . g is ext-real complex real set
T . g is ext-real complex real Element of the carrier of R^1
r . g is ext-real complex real Element of the carrier of R^1
- (r . g) is ext-real complex real set
f . g is ext-real complex real Element of the carrier of R^1
m is non empty TopSpace-like TopStruct
the carrier of m is non empty set
[: the carrier of m, the carrier of R^1:] is Relation-like complex-valued ext-real-valued real-valued set
bool [: the carrier of m, the carrier of R^1:] is set
r is Relation-like non-empty the carrier of m -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of m, the carrier of R^1) complex-valued ext-real-valued real-valued continuous Element of bool [: the carrier of m, the carrier of R^1:]
r " is Relation-like the carrier of m -defined REAL -valued Function-like non empty total V18( the carrier of m, REAL ) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m,REAL:]
[: the carrier of m,REAL:] is Relation-like non trivial complex-valued ext-real-valued real-valued non finite set
bool [: the carrier of m,REAL:] is non trivial non finite set
T is Relation-like the carrier of m -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of m, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m, the carrier of R^1:]
dom r is non empty Element of bool the carrier of m
bool the carrier of m is set
f is Element of the carrier of m
r . f is ext-real complex real Element of the carrier of R^1
f is Relation-like the carrier of m -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of m, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m, the carrier of R^1:]
g is Element of the carrier of m
T . g is ext-real complex real set
f . g is ext-real complex real set
T . g is ext-real complex real Element of the carrier of R^1
r . g is ext-real complex real Element of the carrier of R^1
1 / (r . g) is ext-real complex real Element of COMPLEX
(r . g) " is ext-real complex real set
1 * ((r . g) ") is ext-real complex real set
f . g is ext-real complex real Element of the carrier of R^1
m is non empty TopSpace-like TopStruct
the carrier of m is non empty set
[: the carrier of m, the carrier of R^1:] is Relation-like complex-valued ext-real-valued real-valued set
bool [: the carrier of m, the carrier of R^1:] is set
r is Relation-like the carrier of m -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of m, the carrier of R^1) complex-valued ext-real-valued real-valued continuous Element of bool [: the carrier of m, the carrier of R^1:]
T is ext-real complex real set
T + r is Relation-like the carrier of m -defined Function-like total complex-valued ext-real-valued real-valued set
f is Relation-like the carrier of m -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of m, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m, the carrier of R^1:]
g is Relation-like the carrier of m -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of m, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m, the carrier of R^1:]
p is Element of the carrier of m
f . p is ext-real complex real set
g . p is ext-real complex real set
f . p is ext-real complex real Element of the carrier of R^1
r . p is ext-real complex real Element of the carrier of R^1
(r . p) + T is ext-real complex real set
g . p is ext-real complex real Element of the carrier of R^1
r - T is Relation-like the carrier of m -defined REAL -valued Function-like non empty total V18( the carrier of m, REAL ) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m,REAL:]
[: the carrier of m,REAL:] is Relation-like non trivial complex-valued ext-real-valued real-valued non finite set
bool [: the carrier of m,REAL:] is non trivial non finite set
- T is ext-real complex real set
(- T) + r is Relation-like the carrier of m -defined Function-like total complex-valued ext-real-valued real-valued set
f is Relation-like the carrier of m -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of m, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m, the carrier of R^1:]
T (#) r is Relation-like the carrier of m -defined Function-like total complex-valued ext-real-valued real-valued set
f is Relation-like the carrier of m -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of m, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m, the carrier of R^1:]
g is Relation-like the carrier of m -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of m, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m, the carrier of R^1:]
p is Element of the carrier of m
f . p is ext-real complex real set
g . p is ext-real complex real set
f . p is ext-real complex real Element of the carrier of R^1
r . p is ext-real complex real Element of the carrier of R^1
(r . p) * T is ext-real complex real set
g . p is ext-real complex real Element of the carrier of R^1
r (/) T is Relation-like Function-like complex-valued ext-real-valued real-valued set
1 / T is ext-real complex real set
T " is ext-real complex real set
1 * (T ") is ext-real complex real set
(1 / T) (#) r is Relation-like the carrier of m -defined Function-like total complex-valued ext-real-valued real-valued set
f is Relation-like the carrier of m -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of m, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m, the carrier of R^1:]
m is non empty TopSpace-like TopStruct
the carrier of m is non empty set
[: the carrier of m, the carrier of R^1:] is Relation-like complex-valued ext-real-valued real-valued set
bool [: the carrier of m, the carrier of R^1:] is set
r is Relation-like the carrier of m -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of m, the carrier of R^1) complex-valued ext-real-valued real-valued continuous Element of bool [: the carrier of m, the carrier of R^1:]
T is Relation-like the carrier of m -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of m, the carrier of R^1) complex-valued ext-real-valued real-valued continuous Element of bool [: the carrier of m, the carrier of R^1:]
r + T is Relation-like the carrier of m -defined REAL -valued Function-like non empty total V18( the carrier of m, REAL ) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m,REAL:]
[: the carrier of m,REAL:] is Relation-like non trivial complex-valued ext-real-valued real-valued non finite set
bool [: the carrier of m,REAL:] is non trivial non finite set
f is Relation-like the carrier of m -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of m, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m, the carrier of R^1:]
g is Relation-like the carrier of m -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of m, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m, the carrier of R^1:]
p is Element of the carrier of m
f . p is ext-real complex real set
g . p is ext-real complex real set
f . p is ext-real complex real Element of the carrier of R^1
r . p is ext-real complex real Element of the carrier of R^1
T . p is ext-real complex real Element of the carrier of R^1
(r . p) + (T . p) is ext-real complex real set
g . p is ext-real complex real Element of the carrier of R^1
r - T is Relation-like the carrier of m -defined REAL -valued Function-like non empty total V18( the carrier of m, REAL ) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m,REAL:]
- T is Relation-like the carrier of m -defined Function-like total complex-valued ext-real-valued real-valued set
(- 1) (#) T is Relation-like the carrier of m -defined Function-like total complex-valued ext-real-valued real-valued set
r + (- T) is Relation-like the carrier of m -defined Function-like total complex-valued ext-real-valued real-valued set
- T is Relation-like the carrier of m -defined REAL -valued Function-like non empty total V18( the carrier of m, REAL ) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m,REAL:]
r + (- T) is Relation-like the carrier of m -defined REAL -valued Function-like non empty total V18( the carrier of m, REAL ) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m,REAL:]
f is Relation-like the carrier of m -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of m, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m, the carrier of R^1:]
r (#) T is Relation-like the carrier of m -defined REAL -valued Function-like non empty total V18( the carrier of m, REAL ) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m,REAL:]
f is Relation-like the carrier of m -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of m, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m, the carrier of R^1:]
g is Relation-like the carrier of m -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of m, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m, the carrier of R^1:]
p is Element of the carrier of m
f . p is ext-real complex real set
g . p is ext-real complex real set
f . p is ext-real complex real Element of the carrier of R^1
r . p is ext-real complex real Element of the carrier of R^1
T . p is ext-real complex real Element of the carrier of R^1
(r . p) * (T . p) is ext-real complex real set
g . p is ext-real complex real Element of the carrier of R^1
m is non empty TopSpace-like TopStruct
the carrier of m is non empty set
[: the carrier of m, the carrier of R^1:] is Relation-like complex-valued ext-real-valued real-valued set
bool [: the carrier of m, the carrier of R^1:] is set
r is Relation-like the carrier of m -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of m, the carrier of R^1) complex-valued ext-real-valued real-valued continuous Element of bool [: the carrier of m, the carrier of R^1:]
T is Relation-like non-empty the carrier of m -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of m, the carrier of R^1) complex-valued ext-real-valued real-valued continuous Element of bool [: the carrier of m, the carrier of R^1:]
r /" T is Relation-like the carrier of m -defined REAL -valued Function-like non empty total V18( the carrier of m, REAL ) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m,REAL:]
[: the carrier of m,REAL:] is Relation-like non trivial complex-valued ext-real-valued real-valued non finite set
bool [: the carrier of m,REAL:] is non trivial non finite set
T " is Relation-like the carrier of m -defined Function-like total complex-valued ext-real-valued real-valued set
r (#) (T ") is Relation-like the carrier of m -defined Function-like total complex-valued ext-real-valued real-valued set
T " is Relation-like the carrier of m -defined REAL -valued Function-like non empty total V18( the carrier of m, REAL ) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m,REAL:]
r (#) (T ") is Relation-like the carrier of m -defined REAL -valued Function-like non empty total V18( the carrier of m, REAL ) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m,REAL:]
f is Relation-like the carrier of m -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of m, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of m, the carrier of R^1:]
r is non empty TopSpace-like TopStruct
the carrier of r is non empty set
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of r, the carrier of (TOP-REAL m):] is Relation-like set
bool [: the carrier of r, the carrier of (TOP-REAL m):] is set
T is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued continuous Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
f is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued continuous Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
T <++> f is Relation-like the carrier of r /\ the carrier of r -defined R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m))) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):]
the carrier of r /\ the carrier of r is set
DOMS the carrier of (TOP-REAL m) is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL m) : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL m) : verum } is set
(DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)) is set
R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m))) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):] is Relation-like set
bool [:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):] is set
g is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
p is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
dom g is non empty Element of bool the carrier of r
bool the carrier of r is set
r is Element of the carrier of r
p . r is Relation-like Function-like complex-valued ext-real-valued real-valued set
g . r is Relation-like Function-like complex-valued ext-real-valued real-valued set
p . r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
T . r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
f . r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(T . r) + (f . r) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(T . r) + (f . r) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K855() .: ((T . r),(f . r)) is Relation-like Function-like set
<:(T . r),(f . r):> is Relation-like Function-like set
<:(T . r),(f . r):> (#) K855() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
g . r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
T <--> f is Relation-like the carrier of r /\ the carrier of r -defined R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m))) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):]
g is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
p is Element of the carrier of r
g . p is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
T . p is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
f . p is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(T . p) - (f . p) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- (f . p) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- (f . p) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(- 1) (#) (f . p) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K260((- 1)) is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K857() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined Function-like constant non empty total complex-valued ext-real-valued real-valued V315() V316() set
{(- 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(dom (id REAL)),{(- 1)}:] is Relation-like complex-valued ext-real-valued real-valued set
<:((dom (id REAL)) --> (- 1)),(id REAL):> is Relation-like Function-like set
<:((dom (id REAL)) --> (- 1)),(id REAL):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(f . p) (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
K853() is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
(f . p) (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
(T . p) + (- (f . p)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(T . p) + (- (f . p)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K855() .: ((T . p),(- (f . p))) is Relation-like Function-like set
(T . p) - (f . p) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
- (f . p) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
(T . p) + (- (f . p)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K855() .: ((T . p),(- (f . p))) is Relation-like Function-like set
K856() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K342(REAL,K855(),(id REAL),K853()) is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K856() .: ((T . p),(f . p)) is Relation-like Function-like set
<:(T . p),(f . p):> is Relation-like Function-like set
<:(T . p),(f . p):> (#) K856() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
dom g is non empty Element of bool the carrier of r
bool the carrier of r is set
bool the carrier of (TOP-REAL m) is set
bool the carrier of r is set
p is Element of the carrier of r
g . p is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
r is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL m)
Int r is functional complex-functions-membered ext-real-functions-membered real-functions-membered open Element of bool the carrier of (TOP-REAL m)
Euclid m is non empty V138() V139() strict Reflexive discerning V193() triangle () () MetrStruct
REAL m is functional non empty FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered FinSequenceSet of REAL
m -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V268() Element of REAL * : len b₁ = m } is set
Pitag_dist m is Relation-like [:(REAL m),(REAL m):] -defined REAL -valued Function-like total V18([:(REAL m),(REAL m):], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:(REAL m),(REAL m):],REAL:]
[:(REAL m),(REAL m):] is Relation-like set
[:[:(REAL m),(REAL m):],REAL:] is Relation-like complex-valued ext-real-valued real-valued set
bool [:[:(REAL m),(REAL m):],REAL:] is set
MetrStruct(# (REAL m),(Pitag_dist m) #) is strict MetrStruct
the carrier of (Euclid m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (Euclid m)
W is ext-real complex real set
Ball (q,W) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Euclid m)
bool the carrier of (Euclid m) is set
T . p is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
W1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (Euclid m)
W / 2 is ext-real complex real Element of COMPLEX
W * (2 ") is ext-real complex real set
Ball (W1,(W / 2)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Euclid m)
y is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL m)
the topology of (TOP-REAL m) is non empty Element of bool (bool the carrier of (TOP-REAL m))
bool (bool the carrier of (TOP-REAL m)) is set
TopStruct(# the carrier of (TOP-REAL m), the topology of (TOP-REAL m) #) is non empty strict TopSpace-like TopStruct
TopSpaceMetr (Euclid m) is TopStruct
Family_open_set (Euclid m) is Element of bool (bool the carrier of (Euclid m))
bool (bool the carrier of (Euclid m)) is set
TopStruct(# the carrier of (Euclid m),(Family_open_set (Euclid m)) #) is non empty strict TopStruct
x is Element of bool the carrier of r
T .: x is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL m)
f . p is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
r02 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (Euclid m)
Ball (r02,(W / 2)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Euclid m)
G2 is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL m)
W2 is Element of bool the carrier of r
f .: W2 is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL m)
x /\ W2 is Element of bool the carrier of r
W is Element of bool the carrier of r
g .: W is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL m)
x is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
dom g is non empty Element of bool the carrier of r
z is set
g . z is Relation-like Function-like complex-valued ext-real-valued real-valued set
dom T is non empty Element of bool the carrier of r
pz is Element of the carrier of r
T . pz is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
bb1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (Euclid m)
dist (W1,bb1) is ext-real complex real Element of REAL
the distance of (Euclid m) is Relation-like [: the carrier of (Euclid m), the carrier of (Euclid m):] -defined REAL -valued Function-like total V18([: the carrier of (Euclid m), the carrier of (Euclid m):], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[: the carrier of (Euclid m), the carrier of (Euclid m):],REAL:]
[: the carrier of (Euclid m), the carrier of (Euclid m):] is Relation-like set
[:[: the carrier of (Euclid m), the carrier of (Euclid m):],REAL:] is Relation-like complex-valued ext-real-valued real-valued set
bool [:[: the carrier of (Euclid m), the carrier of (Euclid m):],REAL:] is set
the distance of (Euclid m) . (W1,bb1) is ext-real complex real Element of REAL
[W1,bb1] is set
{W1,bb1} is functional non empty finite V49() complex-functions-membered ext-real-functions-membered real-functions-membered set
{W1} is functional non empty trivial finite V49() 1 -element complex-functions-membered ext-real-functions-membered real-functions-membered set
{{W1,bb1},{W1}} is non empty finite V49() set
the distance of (Euclid m) . [W1,bb1] is ext-real complex real set
(T . p) - (T . pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- (T . pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- (T . pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(- 1) (#) (T . pz) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K260((- 1)) is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K857() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined Function-like constant non empty total complex-valued ext-real-valued real-valued V315() V316() set
{(- 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(dom (id REAL)),{(- 1)}:] is Relation-like complex-valued ext-real-valued real-valued set
<:((dom (id REAL)) --> (- 1)),(id REAL):> is Relation-like Function-like set
<:((dom (id REAL)) --> (- 1)),(id REAL):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(T . pz) (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
K853() is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
(T . pz) (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
(T . p) + (- (T . pz)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(T . p) + (- (T . pz)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K855() .: ((T . p),(- (T . pz))) is Relation-like Function-like set
(T . p) - (T . pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
- (T . pz) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
(T . p) + (- (T . pz)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K855() .: ((T . p),(- (T . pz))) is Relation-like Function-like set
K856() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K342(REAL,K855(),(id REAL),K853()) is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K856() .: ((T . p),(T . pz)) is Relation-like Function-like set
<:(T . p),(T . pz):> is Relation-like Function-like set
<:(T . p),(T . pz):> (#) K856() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
|.((T . p) - (T . pz)).| is ext-real non negative complex real Element of REAL
sqr ((T . p) - (T . pz)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
((T . p) - (T . pz)) (#) ((T . p) - (T . pz)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K857() .: (((T . p) - (T . pz)),((T . p) - (T . pz))) is Relation-like Function-like set
<:((T . p) - (T . pz)),((T . p) - (T . pz)):> is Relation-like Function-like set
<:((T . p) - (T . pz)),((T . p) - (T . pz)):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
((T . p) - (T . pz)) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum (sqr ((T . p) - (T . pz))) is ext-real complex real Element of REAL
sqrt (Sum (sqr ((T . p) - (T . pz)))) is ext-real complex real Element of REAL
dom f is non empty Element of bool the carrier of r
f . pz is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
bb2 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (Euclid m)
dist (r02,bb2) is ext-real complex real Element of REAL
the distance of (Euclid m) . (r02,bb2) is ext-real complex real Element of REAL
[r02,bb2] is set
{r02,bb2} is functional non empty finite V49() complex-functions-membered ext-real-functions-membered real-functions-membered set
{r02} is functional non empty trivial finite V49() 1 -element complex-functions-membered ext-real-functions-membered real-functions-membered set
{{r02,bb2},{r02}} is non empty finite V49() set
the distance of (Euclid m) . [r02,bb2] is ext-real complex real set
(f . p) - (f . pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- (f . pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- (f . pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(- 1) (#) (f . pz) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
(f . pz) (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
(f . pz) (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
(f . p) + (- (f . pz)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(f . p) + (- (f . pz)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() .: ((f . p),(- (f . pz))) is Relation-like Function-like set
(f . p) - (f . pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
- (f . pz) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
(f . p) + (- (f . pz)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K855() .: ((f . p),(- (f . pz))) is Relation-like Function-like set
K856() .: ((f . p),(f . pz)) is Relation-like Function-like set
<:(f . p),(f . pz):> is Relation-like Function-like set
<:(f . p),(f . pz):> (#) K856() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
|.((f . p) - (f . pz)).| is ext-real non negative complex real Element of REAL
sqr ((f . p) - (f . pz)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
((f . p) - (f . pz)) (#) ((f . p) - (f . pz)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K857() .: (((f . p) - (f . pz)),((f . p) - (f . pz))) is Relation-like Function-like set
<:((f . p) - (f . pz)),((f . p) - (f . pz)):> is Relation-like Function-like set
<:((f . p) - (f . pz)),((f . p) - (f . pz)):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
((f . p) - (f . pz)) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum (sqr ((f . p) - (f . pz))) is ext-real complex real Element of REAL
sqrt (Sum (sqr ((f . p) - (f . pz)))) is ext-real complex real Element of REAL
(T . pz) - (f . pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(T . pz) + (- (f . pz)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(T . pz) + (- (f . pz)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() .: ((T . pz),(- (f . pz))) is Relation-like Function-like set
(T . pz) - (f . pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(T . pz) + (- (f . pz)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K855() .: ((T . pz),(- (f . pz))) is Relation-like Function-like set
K856() .: ((T . pz),(f . pz)) is Relation-like Function-like set
<:(T . pz),(f . pz):> is Relation-like Function-like set
<:(T . pz),(f . pz):> (#) K856() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
g . pz is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(T . p) - (f . p) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- (f . p) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- (f . p) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(- 1) (#) (f . p) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
(f . p) (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
(f . p) (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
(T . p) + (- (f . p)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(T . p) + (- (f . p)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() .: ((T . p),(- (f . p))) is Relation-like Function-like set
(T . p) - (f . p) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
- (f . p) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
(T . p) + (- (f . p)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K855() .: ((T . p),(- (f . p))) is Relation-like Function-like set
K856() .: ((T . p),(f . p)) is Relation-like Function-like set
<:(T . p),(f . p):> is Relation-like Function-like set
<:(T . p),(f . p):> (#) K856() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
((T . p) - (f . p)) - ((T . pz) - (f . pz)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- ((T . pz) - (f . pz)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- ((T . pz) - (f . pz)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(- 1) (#) ((T . pz) - (f . pz)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
((T . pz) - (f . pz)) (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
((T . pz) - (f . pz)) (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
((T . p) - (f . p)) + (- ((T . pz) - (f . pz))) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
((T . p) - (f . p)) + (- ((T . pz) - (f . pz))) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() .: (((T . p) - (f . p)),(- ((T . pz) - (f . pz)))) is Relation-like Function-like set
((T . p) - (f . p)) - ((T . pz) - (f . pz)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
- ((T . pz) - (f . pz)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
((T . p) - (f . p)) + (- ((T . pz) - (f . pz))) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K855() .: (((T . p) - (f . p)),(- ((T . pz) - (f . pz)))) is Relation-like Function-like set
K856() .: (((T . p) - (f . p)),((T . pz) - (f . pz))) is Relation-like Function-like set
<:((T . p) - (f . p)),((T . pz) - (f . pz)):> is Relation-like Function-like set
<:((T . p) - (f . p)),((T . pz) - (f . pz)):> (#) K856() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
((T . p) - (f . p)) - (T . pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
((T . p) - (f . p)) + (- (T . pz)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
((T . p) - (f . p)) + (- (T . pz)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() .: (((T . p) - (f . p)),(- (T . pz))) is Relation-like Function-like set
((T . p) - (f . p)) - (T . pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
((T . p) - (f . p)) + (- (T . pz)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K855() .: (((T . p) - (f . p)),(- (T . pz))) is Relation-like Function-like set
K856() .: (((T . p) - (f . p)),(T . pz)) is Relation-like Function-like set
<:((T . p) - (f . p)),(T . pz):> is Relation-like Function-like set
<:((T . p) - (f . p)),(T . pz):> (#) K856() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(((T . p) - (f . p)) - (T . pz)) + (f . pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(((T . p) - (f . p)) - (T . pz)) + (f . pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() .: ((((T . p) - (f . p)) - (T . pz)),(f . pz)) is Relation-like Function-like set
<:(((T . p) - (f . p)) - (T . pz)),(f . pz):> is Relation-like Function-like set
<:(((T . p) - (f . p)) - (T . pz)),(f . pz):> (#) K855() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(T . p) + (- (f . p)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
((T . p) + (- (f . p))) + (- (T . pz)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
((T . p) + (- (f . p))) + (- (T . pz)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() .: (((T . p) + (- (f . p))),(- (T . pz))) is Relation-like Function-like set
<:((T . p) + (- (f . p))),(- (T . pz)):> is Relation-like Function-like set
<:((T . p) + (- (f . p))),(- (T . pz)):> (#) K855() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(((T . p) + (- (f . p))) + (- (T . pz))) + (f . pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(((T . p) + (- (f . p))) + (- (T . pz))) + (f . pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() .: ((((T . p) + (- (f . p))) + (- (T . pz))),(f . pz)) is Relation-like Function-like set
<:(((T . p) + (- (f . p))) + (- (T . pz))),(f . pz):> is Relation-like Function-like set
<:(((T . p) + (- (f . p))) + (- (T . pz))),(f . pz):> (#) K855() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(T . p) + (- (T . pz)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
((T . p) + (- (T . pz))) + (- (f . p)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
((T . p) + (- (T . pz))) + (- (f . p)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() .: (((T . p) + (- (T . pz))),(- (f . p))) is Relation-like Function-like set
<:((T . p) + (- (T . pz))),(- (f . p)):> is Relation-like Function-like set
<:((T . p) + (- (T . pz))),(- (f . p)):> (#) K855() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(((T . p) + (- (T . pz))) + (- (f . p))) + (f . pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(((T . p) + (- (T . pz))) + (- (f . p))) + (f . pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() .: ((((T . p) + (- (T . pz))) + (- (f . p))),(f . pz)) is Relation-like Function-like set
<:(((T . p) + (- (T . pz))) + (- (f . p))),(f . pz):> is Relation-like Function-like set
<:(((T . p) + (- (T . pz))) + (- (f . p))),(f . pz):> (#) K855() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
((T . p) - (T . pz)) - (f . p) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
((T . p) - (T . pz)) + (- (f . p)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
((T . p) - (T . pz)) + (- (f . p)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() .: (((T . p) - (T . pz)),(- (f . p))) is Relation-like Function-like set
((T . p) - (T . pz)) - (f . p) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
((T . p) - (T . pz)) + (- (f . p)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K855() .: (((T . p) - (T . pz)),(- (f . p))) is Relation-like Function-like set
K856() .: (((T . p) - (T . pz)),(f . p)) is Relation-like Function-like set
<:((T . p) - (T . pz)),(f . p):> is Relation-like Function-like set
<:((T . p) - (T . pz)),(f . p):> (#) K856() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(((T . p) - (T . pz)) - (f . p)) + (f . pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(((T . p) - (T . pz)) - (f . p)) + (f . pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() .: ((((T . p) - (T . pz)) - (f . p)),(f . pz)) is Relation-like Function-like set
<:(((T . p) - (T . pz)) - (f . p)),(f . pz):> is Relation-like Function-like set
<:(((T . p) - (T . pz)) - (f . p)),(f . pz):> (#) K855() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
((T . p) - (T . pz)) + (f . pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
((T . p) - (T . pz)) + (f . pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() .: (((T . p) - (T . pz)),(f . pz)) is Relation-like Function-like set
<:((T . p) - (T . pz)),(f . pz):> is Relation-like Function-like set
<:((T . p) - (T . pz)),(f . pz):> (#) K855() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(((T . p) - (T . pz)) + (f . pz)) + (- (f . p)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(((T . p) - (T . pz)) + (f . pz)) + (- (f . p)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() .: ((((T . p) - (T . pz)) + (f . pz)),(- (f . p))) is Relation-like Function-like set
<:(((T . p) - (T . pz)) + (f . pz)),(- (f . p)):> is Relation-like Function-like set
<:(((T . p) - (T . pz)) + (f . pz)),(- (f . p)):> (#) K855() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(f . pz) - (f . p) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(f . pz) + (- (f . p)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(f . pz) + (- (f . p)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() .: ((f . pz),(- (f . p))) is Relation-like Function-like set
(f . pz) - (f . p) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(f . pz) + (- (f . p)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K855() .: ((f . pz),(- (f . p))) is Relation-like Function-like set
K856() .: ((f . pz),(f . p)) is Relation-like Function-like set
<:(f . pz),(f . p):> is Relation-like Function-like set
<:(f . pz),(f . p):> (#) K856() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
((T . p) - (T . pz)) + ((f . pz) - (f . p)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
((T . p) - (T . pz)) + ((f . pz) - (f . p)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() .: (((T . p) - (T . pz)),((f . pz) - (f . p))) is Relation-like Function-like set
<:((T . p) - (T . pz)),((f . pz) - (f . p)):> is Relation-like Function-like set
<:((T . p) - (T . pz)),((f . pz) - (f . p)):> (#) K855() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
|.(((T . p) - (f . p)) - ((T . pz) - (f . pz))).| is ext-real non negative complex real Element of REAL
sqr (((T . p) - (f . p)) - ((T . pz) - (f . pz))) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(((T . p) - (f . p)) - ((T . pz) - (f . pz))) (#) (((T . p) - (f . p)) - ((T . pz) - (f . pz))) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K857() .: ((((T . p) - (f . p)) - ((T . pz) - (f . pz))),(((T . p) - (f . p)) - ((T . pz) - (f . pz)))) is Relation-like Function-like set
<:(((T . p) - (f . p)) - ((T . pz) - (f . pz))),(((T . p) - (f . p)) - ((T . pz) - (f . pz))):> is Relation-like Function-like set
<:(((T . p) - (f . p)) - ((T . pz) - (f . pz))),(((T . p) - (f . p)) - ((T . pz) - (f . pz))):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(((T . p) - (f . p)) - ((T . pz) - (f . pz))) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum (sqr (((T . p) - (f . p)) - ((T . pz) - (f . pz)))) is ext-real complex real Element of REAL
sqrt (Sum (sqr (((T . p) - (f . p)) - ((T . pz) - (f . pz))))) is ext-real complex real Element of REAL
|.((f . pz) - (f . p)).| is ext-real non negative complex real Element of REAL
sqr ((f . pz) - (f . p)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
((f . pz) - (f . p)) (#) ((f . pz) - (f . p)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K857() .: (((f . pz) - (f . p)),((f . pz) - (f . p))) is Relation-like Function-like set
<:((f . pz) - (f . p)),((f . pz) - (f . p)):> is Relation-like Function-like set
<:((f . pz) - (f . p)),((f . pz) - (f . p)):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
((f . pz) - (f . p)) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum (sqr ((f . pz) - (f . p))) is ext-real complex real Element of REAL
sqrt (Sum (sqr ((f . pz) - (f . p)))) is ext-real complex real Element of REAL
|.((T . p) - (T . pz)).| + |.((f . pz) - (f . p)).| is ext-real non negative complex real set
- ((f . pz) - (f . p)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- ((f . pz) - (f . p)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(- 1) (#) ((f . pz) - (f . p)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
((f . pz) - (f . p)) (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
((f . pz) - (f . p)) (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
|.(- ((f . pz) - (f . p))).| is ext-real non negative complex real Element of REAL
sqr (- ((f . pz) - (f . p))) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(- ((f . pz) - (f . p))) (#) (- ((f . pz) - (f . p))) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K857() .: ((- ((f . pz) - (f . p))),(- ((f . pz) - (f . p)))) is Relation-like Function-like set
<:(- ((f . pz) - (f . p))),(- ((f . pz) - (f . p))):> is Relation-like Function-like set
<:(- ((f . pz) - (f . p))),(- ((f . pz) - (f . p))):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(- ((f . pz) - (f . p))) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum (sqr (- ((f . pz) - (f . p)))) is ext-real complex real Element of REAL
sqrt (Sum (sqr (- ((f . pz) - (f . p))))) is ext-real complex real Element of REAL
(W / 2) + (W / 2) is ext-real complex real Element of COMPLEX
|.((T . p) - (T . pz)).| + |.((f . p) - (f . pz)).| is ext-real non negative complex real set
(g . p) - (g . pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- (g . pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
- (g . pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(- 1) (#) (g . pz) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
(g . pz) (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
(g . pz) (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
(g . p) + (- (g . pz)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
(g . p) + (- (g . pz)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() .: ((g . p),(- (g . pz))) is Relation-like Function-like set
(g . p) - (g . pz) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
- (g . pz) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
(g . p) + (- (g . pz)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K855() .: ((g . p),(- (g . pz))) is Relation-like Function-like set
K856() .: ((g . p),(g . pz)) is Relation-like Function-like set
<:(g . p),(g . pz):> is Relation-like Function-like set
<:(g . p),(g . pz):> (#) K856() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
|.((g . p) - (g . pz)).| is ext-real non negative complex real Element of REAL
sqr ((g . p) - (g . pz)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
((g . p) - (g . pz)) (#) ((g . p) - (g . pz)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() set
K857() .: (((g . p) - (g . pz)),((g . p) - (g . pz))) is Relation-like Function-like set
<:((g . p) - (g . pz)),((g . p) - (g . pz)):> is Relation-like Function-like set
<:((g . p) - (g . pz)),((g . p) - (g . pz)):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
((g . p) - (g . pz)) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum (sqr ((g . p) - (g . pz))) is ext-real complex real Element of REAL
sqrt (Sum (sqr ((g . p) - (g . pz)))) is ext-real complex real Element of REAL
bb0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (Euclid m)
dist (q,bb0) is ext-real complex real Element of REAL
the distance of (Euclid m) . (q,bb0) is ext-real complex real Element of REAL
[q,bb0] is set
{q,bb0} is functional non empty finite V49() complex-functions-membered ext-real-functions-membered real-functions-membered set
{q} is functional non empty trivial finite V49() 1 -element complex-functions-membered ext-real-functions-membered real-functions-membered set
{{q,bb0},{q}} is non empty finite V49() set
the distance of (Euclid m) . [q,bb0] is ext-real complex real set
T <##> f is Relation-like the carrier of r /\ the carrier of r -defined R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m))) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):]
(m) is Relation-like the carrier of [:(TOP-REAL m),(TOP-REAL m):] -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of [:(TOP-REAL m),(TOP-REAL m):], the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued continuous Element of bool [: the carrier of [:(TOP-REAL m),(TOP-REAL m):], the carrier of (TOP-REAL m):]
[:(TOP-REAL m),(TOP-REAL m):] is non empty strict TopSpace-like TopStruct
the carrier of [:(TOP-REAL m),(TOP-REAL m):] is non empty set
[: the carrier of [:(TOP-REAL m),(TOP-REAL m):], the carrier of (TOP-REAL m):] is Relation-like set
bool [: the carrier of [:(TOP-REAL m),(TOP-REAL m):], the carrier of (TOP-REAL m):] is set
(m) .: (T,f) is Relation-like Function-like set
<:T,f:> is Relation-like Function-like set
<:T,f:> (#) (m) is Relation-like the carrier of (TOP-REAL m) -valued Function-like Function-yielding V156() complex-functions-valued set
<:T,f:> is Relation-like the carrier of r -defined [: the carrier of (TOP-REAL m), the carrier of (TOP-REAL m):] -valued Function-like V18( the carrier of r,[: the carrier of (TOP-REAL m), the carrier of (TOP-REAL m):]) Element of bool [: the carrier of r,[: the carrier of (TOP-REAL m), the carrier of (TOP-REAL m):]:]
[: the carrier of (TOP-REAL m), the carrier of (TOP-REAL m):] is Relation-like set
[: the carrier of r,[: the carrier of (TOP-REAL m), the carrier of (TOP-REAL m):]:] is Relation-like set
bool [: the carrier of r,[: the carrier of (TOP-REAL m), the carrier of (TOP-REAL m):]:] is set
[: the carrier of r, the carrier of [:(TOP-REAL m),(TOP-REAL m):]:] is Relation-like set
bool [: the carrier of r, the carrier of [:(TOP-REAL m),(TOP-REAL m):]:] is set
g is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
r is non empty TopSpace-like TopStruct
the carrier of r is non empty set
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of r, the carrier of (TOP-REAL m):] is Relation-like set
bool [: the carrier of r, the carrier of (TOP-REAL m):] is set
[: the carrier of r, the carrier of R^1:] is Relation-like complex-valued ext-real-valued real-valued set
bool [: the carrier of r, the carrier of R^1:] is set
f is Relation-like the carrier of r -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of r, the carrier of R^1) complex-valued ext-real-valued real-valued continuous Element of bool [: the carrier of r, the carrier of R^1:]
(m, the carrier of r, the carrier of R^1,f) is Relation-like the carrier of r -defined the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total total V18( the carrier of r, the carrier of (TOP-REAL m)) V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued continuous Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
T is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued continuous Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
T <+> f is Relation-like the carrier of r /\ (dom f) -defined R_PFuncs (DOMS the carrier of (TOP-REAL m)) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:( the carrier of r /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL m))):]
dom f is non empty set
the carrier of r /\ (dom f) is set
DOMS the carrier of (TOP-REAL m) is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL m) : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL m) : verum } is set
R_PFuncs (DOMS the carrier of (TOP-REAL m)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:( the carrier of r /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL m))):] is Relation-like set
bool [:( the carrier of r /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL m))):] is set
p is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
T <++> (m, the carrier of r, the carrier of R^1,f) is Relation-like the carrier of r /\ the carrier of r -defined R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m))) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):]
the carrier of r /\ the carrier of r is set
(DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)) is set
R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m))) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):] is Relation-like set
bool [:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):] is set
r is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
T <-> f is Relation-like the carrier of r /\ (dom f) -defined R_PFuncs (DOMS the carrier of (TOP-REAL m)) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:( the carrier of r /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL m))):]
- f is Relation-like the carrier of r -defined Function-like total complex-valued ext-real-valued real-valued set
(- 1) (#) f is Relation-like the carrier of r -defined Function-like total complex-valued ext-real-valued real-valued set
T <+> (- f) is Relation-like Function-like set
p is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
T <--> (m, the carrier of r, the carrier of R^1,f) is Relation-like the carrier of r /\ the carrier of r -defined R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m))) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):]
the carrier of r /\ the carrier of r is set
(DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)) is set
R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m))) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):] is Relation-like set
bool [:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):] is set
r is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
T <#> f is Relation-like the carrier of r /\ (dom f) -defined R_PFuncs (DOMS the carrier of (TOP-REAL m)) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:( the carrier of r /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL m))):]
p is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
T <##> (m, the carrier of r, the carrier of R^1,f) is Relation-like the carrier of r /\ the carrier of r -defined R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m))) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):]
the carrier of r /\ the carrier of r is set
(DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)) is set
R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m))) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):] is Relation-like set
bool [:( the carrier of r /\ the carrier of r),(R_PFuncs ((DOMS the carrier of (TOP-REAL m)) /\ (DOMS the carrier of (TOP-REAL m)))):] is set
r is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
r is non empty TopSpace-like TopStruct
the carrier of r is non empty set
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of r, the carrier of (TOP-REAL m):] is Relation-like set
bool [: the carrier of r, the carrier of (TOP-REAL m):] is set
[: the carrier of r, the carrier of R^1:] is Relation-like complex-valued ext-real-valued real-valued set
bool [: the carrier of r, the carrier of R^1:] is set
T is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued continuous Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
f is Relation-like non-empty the carrier of r -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of r, the carrier of R^1) complex-valued ext-real-valued real-valued continuous Element of bool [: the carrier of r, the carrier of R^1:]
T </> f is Relation-like the carrier of r /\ (dom f) -defined R_PFuncs (DOMS the carrier of (TOP-REAL m)) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:( the carrier of r /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL m))):]
dom f is non empty set
the carrier of r /\ (dom f) is set
DOMS the carrier of (TOP-REAL m) is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL m) : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL m) : verum } is set
R_PFuncs (DOMS the carrier of (TOP-REAL m)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[:( the carrier of r /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL m))):] is Relation-like set
bool [:( the carrier of r /\ (dom f)),(R_PFuncs (DOMS the carrier of (TOP-REAL m))):] is set
f " is Relation-like the carrier of r -defined Function-like total complex-valued ext-real-valued real-valued set
T <#> (f ") is Relation-like Function-like set
g is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
f " is Relation-like the carrier of r -defined REAL -valued Function-like non empty total V18( the carrier of r, REAL ) complex-valued ext-real-valued real-valued Element of bool [: the carrier of r,REAL:]
[: the carrier of r,REAL:] is Relation-like non trivial complex-valued ext-real-valued real-valued non finite set
bool [: the carrier of r,REAL:] is non trivial non finite set
p is Relation-like the carrier of r -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of r, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of r, the carrier of R^1:]
r is non empty TopSpace-like TopStruct
the carrier of r is non empty set
m is ext-real non negative ordinal natural complex real finite cardinal set
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of r, the carrier of (TOP-REAL m):] is Relation-like set
bool [: the carrier of r, the carrier of (TOP-REAL m):] is set
f is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued continuous Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
T is ext-real complex real set
f [+] T is Relation-like the carrier of r -defined R_PFuncs (DOMS the carrier of (TOP-REAL m)) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r,(R_PFuncs (DOMS the carrier of (TOP-REAL m))):]
DOMS the carrier of (TOP-REAL m) is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL m) : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of the carrier of (TOP-REAL m) : verum } is set
R_PFuncs (DOMS the carrier of (TOP-REAL m)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of r,(R_PFuncs (DOMS the carrier of (TOP-REAL m))):] is Relation-like set
bool [: the carrier of r,(R_PFuncs (DOMS the carrier of (TOP-REAL m))):] is set
g is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
dom f is non empty Element of bool the carrier of r
bool the carrier of r is set
p is ext-real complex real Element of the carrier of R^1
r --> p is Relation-like the carrier of r -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of r, the carrier of R^1) complex-valued ext-real-valued real-valued continuous Element of bool [: the carrier of r, the carrier of R^1:]
[: the carrier of r, the carrier of R^1:] is Relation-like complex-valued ext-real-valued real-valued set
bool [: the carrier of r, the carrier of R^1:] is set
the carrier of r --> p is Relation-like the carrier of r -defined the carrier of R^1 -valued Function-like constant non empty total V18( the carrier of r, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of r, the carrier of R^1:]
{p} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[: the carrier of r,{p}:] is Relation-like complex-valued ext-real-valued real-valued set
f <+> (r --> p) is Relation-like the carrier of r /\ (dom (r --> p)) -defined R_PFuncs (DOMS the carrier of (TOP-REAL m)) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:( the carrier of r /\ (dom (r --> p))),(R_PFuncs (DOMS the carrier of (TOP-REAL m))):]
dom (r --> p) is non empty set
the carrier of r /\ (dom (r --> p)) is set
[:( the carrier of r /\ (dom (r --> p))),(R_PFuncs (DOMS the carrier of (TOP-REAL m))):] is Relation-like set
bool [:( the carrier of r /\ (dom (r --> p))),(R_PFuncs (DOMS the carrier of (TOP-REAL m))):] is set
f [-] T is Relation-like the carrier of r -defined R_PFuncs (DOMS the carrier of (TOP-REAL m)) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r,(R_PFuncs (DOMS the carrier of (TOP-REAL m))):]
- T is ext-real complex real set
f [+] (- T) is Relation-like Function-like set
g is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
f [#] T is Relation-like the carrier of r -defined R_PFuncs (DOMS the carrier of (TOP-REAL m)) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r,(R_PFuncs (DOMS the carrier of (TOP-REAL m))):]
g is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
dom f is non empty Element of bool the carrier of r
bool the carrier of r is set
p is ext-real complex real Element of the carrier of R^1
r --> p is Relation-like the carrier of r -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of r, the carrier of R^1) complex-valued ext-real-valued real-valued continuous Element of bool [: the carrier of r, the carrier of R^1:]
[: the carrier of r, the carrier of R^1:] is Relation-like complex-valued ext-real-valued real-valued set
bool [: the carrier of r, the carrier of R^1:] is set
the carrier of r --> p is Relation-like the carrier of r -defined the carrier of R^1 -valued Function-like constant non empty total V18( the carrier of r, the carrier of R^1) complex-valued ext-real-valued real-valued Element of bool [: the carrier of r, the carrier of R^1:]
{p} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[: the carrier of r,{p}:] is Relation-like complex-valued ext-real-valued real-valued set
f <#> (r --> p) is Relation-like the carrier of r /\ (dom (r --> p)) -defined R_PFuncs (DOMS the carrier of (TOP-REAL m)) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [:( the carrier of r /\ (dom (r --> p))),(R_PFuncs (DOMS the carrier of (TOP-REAL m))):]
dom (r --> p) is non empty set
the carrier of r /\ (dom (r --> p)) is set
[:( the carrier of r /\ (dom (r --> p))),(R_PFuncs (DOMS the carrier of (TOP-REAL m))):] is Relation-like set
bool [:( the carrier of r /\ (dom (r --> p))),(R_PFuncs (DOMS the carrier of (TOP-REAL m))):] is set
f [/] T is Relation-like the carrier of r -defined R_PFuncs (DOMS the carrier of (TOP-REAL m)) -valued Function-like Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r,(R_PFuncs (DOMS the carrier of (TOP-REAL m))):]
T " is ext-real complex real set
f [#] (T ") is Relation-like Function-like set
g is Relation-like the carrier of r -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of r, the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of r, the carrier of (TOP-REAL m):]
m is ext-real non negative complex real set
r is non empty ext-real positive non negative ordinal natural complex real finite cardinal set
TOP-REAL r is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
0. (TOP-REAL r) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like zero V268() Element of the carrier of (TOP-REAL r)
the carrier of (TOP-REAL r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
the ZeroF of (TOP-REAL r) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
Tcircle ((0. (TOP-REAL r)),m) is non empty strict TopSpace-like () () SubSpace of TOP-REAL r
Sphere ((0. (TOP-REAL r)),m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered closed Element of bool the carrier of (TOP-REAL r)
bool the carrier of (TOP-REAL r) is set
(TOP-REAL r) | (Sphere ((0. (TOP-REAL r)),m)) is non empty strict TopSpace-like () () SubSpace of TOP-REAL r
the carrier of (Tcircle ((0. (TOP-REAL r)),m)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
T is Relation-like Function-like complex-valued ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL r)),m))
- T is Relation-like Function-like complex-valued ext-real-valued real-valued set
(- 1) (#) T is Relation-like Function-like complex-valued ext-real-valued real-valued set
f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
- f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
- f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(- 1) (#) f is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
K260((- 1)) is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K857() [;] ((- 1),(id REAL)) is Relation-like Function-like set
(dom (id REAL)) --> (- 1) is Relation-like non-empty dom (id REAL) -defined Function-like constant non empty total complex-valued ext-real-valued real-valued V315() V316() set
{(- 1)} is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered set
[:(dom (id REAL)),{(- 1)}:] is Relation-like complex-valued ext-real-valued real-valued set
<:((dom (id REAL)) --> (- 1)),(id REAL):> is Relation-like Function-like set
<:((dom (id REAL)) --> (- 1)),(id REAL):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
f (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
K853() is Relation-like REAL -defined REAL -valued Function-like non empty total V18( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
f (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
(- f) - (0. (TOP-REAL r)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
- (0. (TOP-REAL r)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
- (0. (TOP-REAL r)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(- 1) (#) (0. (TOP-REAL r)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
(0. (TOP-REAL r)) (#) K260((- 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
(0. (TOP-REAL r)) (#) K853() is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
(- f) + (- (0. (TOP-REAL r))) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
(- f) + (- (0. (TOP-REAL r))) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K855() .: ((- f),(- (0. (TOP-REAL r)))) is Relation-like Function-like set
(- f) - (0. (TOP-REAL r)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
- (0. (TOP-REAL r)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
(- f) + (- (0. (TOP-REAL r))) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
K855() .: ((- f),(- (0. (TOP-REAL r)))) is Relation-like Function-like set
K856() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K342(REAL,K855(),(id REAL),K853()) is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total V18([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K856() .: ((- f),(0. (TOP-REAL r))) is Relation-like Function-like set
<:(- f),(0. (TOP-REAL r)):> is Relation-like Function-like set
<:(- f),(0. (TOP-REAL r)):> (#) K856() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
|.((- f) - (0. (TOP-REAL r))).| is ext-real non negative complex real Element of REAL
sqr ((- f) - (0. (TOP-REAL r))) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
((- f) - (0. (TOP-REAL r))) (#) ((- f) - (0. (TOP-REAL r))) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
K857() .: (((- f) - (0. (TOP-REAL r))),((- f) - (0. (TOP-REAL r)))) is Relation-like Function-like set
<:((- f) - (0. (TOP-REAL r))),((- f) - (0. (TOP-REAL r))):> is Relation-like Function-like set
<:((- f) - (0. (TOP-REAL r))),((- f) - (0. (TOP-REAL r))):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
((- f) - (0. (TOP-REAL r))) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum (sqr ((- f) - (0. (TOP-REAL r)))) is ext-real complex real Element of REAL
sqrt (Sum (sqr ((- f) - (0. (TOP-REAL r))))) is ext-real complex real Element of REAL
|.(- f).| is ext-real non negative complex real Element of REAL
sqr (- f) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(- f) (#) (- f) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
K857() .: ((- f),(- f)) is Relation-like Function-like set
<:(- f),(- f):> is Relation-like Function-like set
<:(- f),(- f):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(- f) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum (sqr (- f)) is ext-real complex real Element of REAL
sqrt (Sum (sqr (- f))) is ext-real complex real Element of REAL
|.f.| is ext-real non negative complex real Element of REAL
sqr f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
f (#) f is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
K857() .: (f,f) is Relation-like Function-like set
<:f,f:> is Relation-like Function-like set
<:f,f:> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
f (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum (sqr f) is ext-real complex real Element of REAL
sqrt (Sum (sqr f)) is ext-real complex real Element of REAL
f - (0. (TOP-REAL r)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
f + (- (0. (TOP-REAL r))) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL r)
f + (- (0. (TOP-REAL r))) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
K855() .: (f,(- (0. (TOP-REAL r)))) is Relation-like Function-like set
f - (0. (TOP-REAL r)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
f + (- (0. (TOP-REAL r))) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
K855() .: (f,(- (0. (TOP-REAL r)))) is Relation-like Function-like set
K856() .: (f,(0. (TOP-REAL r))) is Relation-like Function-like set
<:f,(0. (TOP-REAL r)):> is Relation-like Function-like set
<:f,(0. (TOP-REAL r)):> (#) K856() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
|.(f - (0. (TOP-REAL r))).| is ext-real non negative complex real Element of REAL
sqr (f - (0. (TOP-REAL r))) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() FinSequence of REAL
(f - (0. (TOP-REAL r))) (#) (f - (0. (TOP-REAL r))) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued finite r -element FinSequence-like FinSubsequence-like V268() set
K857() .: ((f - (0. (TOP-REAL r))),(f - (0. (TOP-REAL r)))) is Relation-like Function-like set
<:(f - (0. (TOP-REAL r))),(f - (0. (TOP-REAL r))):> is Relation-like Function-like set
<:(f - (0. (TOP-REAL r))),(f - (0. (TOP-REAL r))):> (#) K857() is Relation-like REAL -valued Function-like complex-valued ext-real-valued real-valued set
(f - (0. (TOP-REAL r))) (#) sqrreal is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite FinSequence-like FinSubsequence-like V268() set
Sum (sqr (f - (0. (TOP-REAL r)))) is ext-real complex real Element of REAL
sqrt (Sum (sqr (f - (0. (TOP-REAL r))))) is ext-real complex real Element of REAL
m is ext-real non negative ordinal natural complex real finite cardinal set
m + 1 is non empty ext-real positive non negative ordinal natural complex real finite cardinal set
TOP-REAL (m + 1) is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
0. (TOP-REAL (m + 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m + 1 -element FinSequence-like FinSubsequence-like zero V268() Element of the carrier of (TOP-REAL (m + 1))
the carrier of (TOP-REAL (m + 1)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
the ZeroF of (TOP-REAL (m + 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m + 1 -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL (m + 1))
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
r is ext-real non negative complex real set
Tcircle ((0. (TOP-REAL (m + 1))),r) is non empty strict TopSpace-like () () SubSpace of TOP-REAL (m + 1)
Sphere ((0. (TOP-REAL (m + 1))),r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered closed Element of bool the carrier of (TOP-REAL (m + 1))
bool the carrier of (TOP-REAL (m + 1)) is set
(TOP-REAL (m + 1)) | (Sphere ((0. (TOP-REAL (m + 1))),r)) is non empty strict TopSpace-like () () SubSpace of TOP-REAL (m + 1)
the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)), the carrier of (TOP-REAL m):] is Relation-like set
bool [: the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)), the carrier of (TOP-REAL m):] is set
f is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)), the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)), the carrier of (TOP-REAL m):]
f (-) is Relation-like Function-like set
dom (f (-)) is set
dom f is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r))
bool the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) is set
p is set
(f (-)) . p is set
r is Relation-like Function-like complex-valued ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r))
- r is Relation-like Function-like complex-valued ext-real-valued real-valued set
(- 1) (#) r is Relation-like Function-like complex-valued ext-real-valued real-valued set
(f (-)) . r is set
W is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)), the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)), the carrier of (TOP-REAL m):]
q is Relation-like Function-like complex-valued ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r))
W . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
m is ext-real non negative ordinal natural complex real finite cardinal set
m + 1 is non empty ext-real positive non negative ordinal natural complex real finite cardinal set
TOP-REAL (m + 1) is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
0. (TOP-REAL (m + 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m + 1 -element FinSequence-like FinSubsequence-like zero V268() Element of the carrier of (TOP-REAL (m + 1))
the carrier of (TOP-REAL (m + 1)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
the ZeroF of (TOP-REAL (m + 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m + 1 -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL (m + 1))
r is ext-real non negative complex real set
Tcircle ((0. (TOP-REAL (m + 1))),r) is non empty strict TopSpace-like () () SubSpace of TOP-REAL (m + 1)
Sphere ((0. (TOP-REAL (m + 1))),r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered closed Element of bool the carrier of (TOP-REAL (m + 1))
bool the carrier of (TOP-REAL (m + 1)) is set
(TOP-REAL (m + 1)) | (Sphere ((0. (TOP-REAL (m + 1))),r)) is non empty strict TopSpace-like () () SubSpace of TOP-REAL (m + 1)
the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
bool the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) is set
T is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r))
(T) is functional complex-functions-membered ext-real-functions-membered real-functions-membered set
g is set
p is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (T)
- p is Relation-like Function-like complex-valued ext-real-valued real-valued set
(- 1) (#) p is Relation-like Function-like complex-valued ext-real-valued real-valued set
- (- p) is Relation-like Function-like complex-valued ext-real-valued real-valued set
(- 1) (#) (- p) is Relation-like Function-like complex-valued ext-real-valued real-valued set
m is ext-real non negative ordinal natural complex real finite cardinal set
m + 1 is non empty ext-real positive non negative ordinal natural complex real finite cardinal set
TOP-REAL (m + 1) is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
0. (TOP-REAL (m + 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m + 1 -element FinSequence-like FinSubsequence-like zero V268() Element of the carrier of (TOP-REAL (m + 1))
the carrier of (TOP-REAL (m + 1)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
the ZeroF of (TOP-REAL (m + 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m + 1 -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL (m + 1))
r is ext-real non negative complex real set
Tcircle ((0. (TOP-REAL (m + 1))),r) is non empty strict TopSpace-like () () SubSpace of TOP-REAL (m + 1)
Sphere ((0. (TOP-REAL (m + 1))),r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered closed Element of bool the carrier of (TOP-REAL (m + 1))
bool the carrier of (TOP-REAL (m + 1)) is set
(TOP-REAL (m + 1)) | (Sphere ((0. (TOP-REAL (m + 1))),r)) is non empty strict TopSpace-like () () SubSpace of TOP-REAL (m + 1)
the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
bool the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) is set
T is functional complex-functions-membered ext-real-functions-membered real-functions-membered open Element of bool the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r))
(m,r,T) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r))
g is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL (m + 1))
g /\ the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL (m + 1))
((m + 1),g) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL (m + 1))
((m + 1),g) /\ the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL (m + 1))
p is set
r is Relation-like Function-like complex-valued ext-real-valued real-valued Element of (m,r,T)
- r is Relation-like Function-like complex-valued ext-real-valued real-valued set
(- 1) (#) r is Relation-like Function-like complex-valued ext-real-valued real-valued set
p is set
r is Relation-like Function-like complex-valued ext-real-valued real-valued Element of ((m + 1),g)
- r is Relation-like Function-like complex-valued ext-real-valued real-valued set
(- 1) (#) r is Relation-like Function-like complex-valued ext-real-valued real-valued set
g is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r))
p is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL (m + 1))
p /\ the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL (m + 1))
m is ext-real non negative ordinal natural complex real finite cardinal set
m + 1 is non empty ext-real positive non negative ordinal natural complex real finite cardinal set
TOP-REAL (m + 1) is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
0. (TOP-REAL (m + 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m + 1 -element FinSequence-like FinSubsequence-like zero V268() Element of the carrier of (TOP-REAL (m + 1))
the carrier of (TOP-REAL (m + 1)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
the ZeroF of (TOP-REAL (m + 1)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m + 1 -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL (m + 1))
TOP-REAL m is non empty left_complementable right_complementable V138() V139() TopSpace-like Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital zeroed strict () () L16()
the carrier of (TOP-REAL m) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
r is ext-real non negative complex real set
Tcircle ((0. (TOP-REAL (m + 1))),r) is non empty strict TopSpace-like () () SubSpace of TOP-REAL (m + 1)
Sphere ((0. (TOP-REAL (m + 1))),r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered closed Element of bool the carrier of (TOP-REAL (m + 1))
bool the carrier of (TOP-REAL (m + 1)) is set
(TOP-REAL (m + 1)) | (Sphere ((0. (TOP-REAL (m + 1))),r)) is non empty strict TopSpace-like () () SubSpace of TOP-REAL (m + 1)
the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered set
[: the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)), the carrier of (TOP-REAL m):] is Relation-like set
bool [: the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)), the carrier of (TOP-REAL m):] is set
f is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)), the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued continuous Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)), the carrier of (TOP-REAL m):]
f (-) is Relation-like Function-like set
bool the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) is set
g is Relation-like the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) -defined the carrier of (TOP-REAL m) -valued Function-like non empty total V18( the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)), the carrier of (TOP-REAL m)) Function-yielding V156() complex-functions-valued ext-real-functions-valued real-functions-valued Element of bool [: the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)), the carrier of (TOP-REAL m):]
p is Relation-like Function-like complex-valued ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r))
g . p is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
r is non empty ext-real positive non negative complex real set
Ball ((g . p),r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered open V269( TOP-REAL m) Element of bool the carrier of (TOP-REAL m)
bool the carrier of (TOP-REAL m) is set
- p is Relation-like Function-like complex-valued ext-real-valued real-valued set
(- 1) (#) p is Relation-like Function-like complex-valued ext-real-valued real-valued set
q is Relation-like Function-like complex-valued ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r))
f . q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
Ball ((f . q),r) is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered open V269( TOP-REAL m) Element of bool the carrier of (TOP-REAL m)
W is functional complex-functions-membered ext-real-functions-membered real-functions-membered open Element of bool the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r))
f .: W is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL m)
(m,r,W) is functional complex-functions-membered ext-real-functions-membered real-functions-membered open Element of bool the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r))
W1 is functional complex-functions-membered ext-real-functions-membered real-functions-membered open Element of bool the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r))
g .: W1 is functional complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (TOP-REAL m)
- q is Relation-like Function-like complex-valued ext-real-valued real-valued set
(- 1) (#) q is Relation-like Function-like complex-valued ext-real-valued real-valued set
y is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
x is Relation-like Function-like complex-valued ext-real-valued real-valued Element of the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r))
g . x is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite m -element FinSequence-like FinSubsequence-like V268() Element of the carrier of (TOP-REAL m)
dom g is functional non empty complex-functions-membered ext-real-functions-membered real-functions-membered Element of bool the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r))
- 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 . (- x) is Relation-like Function-like complex-valued ext-real-valued real-valued set
f . (- p) is Relation-like Function-like complex-valued ext-real-valued real-valued set
(m,r,W1) is functional complex-functions-membered ext-real-functions-membered real-functions-membered open Element of bool the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r))