:: TOPGEN_5 semantic presentation

REAL is non empty non trivial non finite V169() V170() V171() V175() V269() V270() V272() set
NAT is ordinal non trivial non finite cardinal limit_cardinal V169() V170() V171() V172() V173() V174() V175() V269() Element of bool REAL
bool REAL is non empty non trivial non finite set
COMPLEX is non empty non trivial non finite V169() V175() set
omega is ordinal non trivial non finite cardinal limit_cardinal V169() V170() V171() V172() V173() V174() V175() V269() set
bool omega is non empty non trivial non finite set
[:NAT,REAL:] is Relation-like non trivial non finite V159() V160() V161() set
bool [:NAT,REAL:] is non empty non trivial non finite set
bool (bool REAL) is non empty non trivial non finite set
bool NAT is non empty non trivial non finite set
RAT is non empty non trivial non finite V169() V170() V171() V172() V175() set
INT is non empty non trivial non finite V169() V170() V171() V172() V173() V175() set
[:REAL,REAL:] is Relation-like non empty non trivial non finite V159() V160() V161() set
bool [:REAL,REAL:] is non empty non trivial non finite set
K408() is non empty V118() L9()
the carrier of K408() is non empty set
K413() is non empty V118() V140() V141() V142() V144() V191() V192() V193() V194() V195() V196() L9()
K414() is non empty V118() V142() V144() V194() V195() V196() M18(K413())
K415() is non empty V118() V140() V142() V144() V194() V195() V196() V197() M21(K413(),K414())
K417() is non empty V118() V140() V142() V144() L9()
K418() is non empty V118() V140() V142() V144() V197() M18(K417())
1 is ordinal natural non empty V28() real ext-real positive non negative finite cardinal V99() rational V169() V170() V171() V172() V173() V174() V267() V268() V269() V270() V271() Element of NAT
[:1,1:] is Relation-like RAT -valued INT -valued non empty finite V159() V160() V161() V162() set
bool [:1,1:] is non empty finite V37() set
[:[:1,1:],1:] is Relation-like RAT -valued INT -valued non empty finite V159() V160() V161() V162() set
bool [:[:1,1:],1:] is non empty finite V37() set
[:[:1,1:],REAL:] is Relation-like non empty non trivial non finite V159() V160() V161() set
bool [:[:1,1:],REAL:] is non empty non trivial non finite set
[:[:REAL,REAL:],REAL:] is Relation-like non empty non trivial non finite V159() V160() V161() set
bool [:[:REAL,REAL:],REAL:] is non empty non trivial non finite set
2 is ordinal natural non empty V28() real ext-real positive non negative finite cardinal V99() rational V169() V170() V171() V172() V173() V174() V267() V268() V269() V270() V271() Element of NAT
[:2,2:] is Relation-like RAT -valued INT -valued non empty finite V159() V160() V161() V162() set
[:[:2,2:],REAL:] is Relation-like non empty non trivial non finite V159() V160() V161() set
bool [:[:2,2:],REAL:] is non empty non trivial non finite set
K525() is non empty strict TopSpace-like V229() TopStruct
the carrier of K525() is non empty V169() V170() V171() set
K454() is V201() V229() L14()
R^1 is non empty strict TopSpace-like V229() TopStruct
I[01] is non empty strict TopSpace-like V229() SubSpace of R^1
TOP-REAL 2 is non empty TopSpace-like V116() V181() V182() V241() V242() V243() V244() V245() V246() V247() V253() L16()
the carrier of (TOP-REAL 2) is non empty set
[:COMPLEX,COMPLEX:] is Relation-like non empty non trivial non finite V159() set
bool [:COMPLEX,COMPLEX:] is non empty non trivial non finite set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is Relation-like non empty non trivial non finite V159() set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty non trivial non finite set
[:RAT,RAT:] is Relation-like RAT -valued non empty non trivial non finite V159() V160() V161() set
bool [:RAT,RAT:] is non empty non trivial non finite set
[:[:RAT,RAT:],RAT:] is Relation-like RAT -valued non empty non trivial non finite V159() V160() V161() set
bool [:[:RAT,RAT:],RAT:] is non empty non trivial non finite set
[:INT,INT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite V159() V160() V161() set
bool [:INT,INT:] is non empty non trivial non finite set
[:[:INT,INT:],INT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite V159() V160() V161() set
bool [:[:INT,INT:],INT:] is non empty non trivial non finite set
[:NAT,NAT:] is Relation-like RAT -valued INT -valued V159() V160() V161() V162() set
[:[:NAT,NAT:],NAT:] is Relation-like RAT -valued INT -valued V159() V160() V161() V162() set
bool [:[:NAT,NAT:],NAT:] is non empty set
{} is set
the Relation-like non-empty empty-yielding NAT -defined RAT -valued ordinal natural Function-like one-to-one constant functional empty V28() real ext-real non positive non negative finite finite-yielding V37() cardinal {} -element Function-yielding V65() FinSequence-like FinSubsequence-like FinSequence-membered V159() V160() V161() V162() V169() V170() V171() V172() V173() V174() V175() V269() V270() V271() V272() set is Relation-like non-empty empty-yielding NAT -defined RAT -valued ordinal natural Function-like one-to-one constant functional empty V28() real ext-real non positive non negative finite finite-yielding V37() cardinal {} -element Function-yielding V65() FinSequence-like FinSubsequence-like FinSequence-membered V159() V160() V161() V162() V169() V170() V171() V172() V173() V174() V175() V269() V270() V271() V272() set
3 is ordinal natural non empty V28() real ext-real positive non negative finite cardinal V99() rational V169() V170() V171() V172() V173() V174() V267() V268() V269() V270() V271() Element of NAT
0 is ordinal natural V28() real ext-real finite cardinal V99() rational V169() V170() V171() V172() V173() V174() V269() V270() V271() Element of NAT
Sorgenfrey-line is non empty strict TopSpace-like TopStruct
{ [.b₁,b₂.[ where b₁, b₂ is V28() real ext-real Element of REAL : ( not b₂ <= b₁ & b₂ is rational ) } is set
the carrier of Sorgenfrey-line is non empty set
bool the carrier of Sorgenfrey-line is non empty set
len RAT is ordinal non empty non trivial non finite cardinal set
continuum is ordinal non trivial non finite cardinal set
len REAL is ordinal non empty non trivial non finite cardinal set
exp (2,omega) is ordinal cardinal set
Seg 1 is non empty trivial finite 1 -element V169() V170() V171() V172() V173() V174() V267() V268() V269() V270() V271() Element of bool NAT
{1} is non empty trivial finite V37() 1 -element V169() V170() V171() V172() V173() V174() V267() V268() V269() V270() V271() set
Seg 2 is non empty finite 2 -element V169() V170() V171() V172() V173() V174() V267() V268() V269() V270() V271() Element of bool NAT
{1,2} is finite V37() V169() V170() V171() V172() V173() V174() V269() V270() V271() set
omega *` omega is ordinal cardinal set
[.0,1.] is V169() V170() V171() V272() Element of bool REAL
bool the carrier of K525() is non empty set
- 1 is non empty V28() real ext-real non positive negative rational Element of REAL
Closed-Interval-TSpace ((- 1),1) is non empty strict TopSpace-like V229() SubSpace of R^1
the carrier of (Closed-Interval-TSpace ((- 1),1)) is non empty V169() V170() V171() set
the carrier of I[01] is non empty V169() V170() V171() set
[: the carrier of I[01], the carrier of (Closed-Interval-TSpace ((- 1),1)):] is Relation-like non empty V159() V160() V161() set
bool [: the carrier of I[01], the carrier of (Closed-Interval-TSpace ((- 1),1)):] is non empty set
the carrier of R^1 is non empty V169() V170() V171() set
proj1 {} is set
proj2 {} is set
bool the carrier of R^1 is non empty set
-infty is non empty non real ext-real non positive negative set
+infty is non empty non real ext-real positive non negative set
K200(-infty,+infty) is V267() V268() V272() set
X is Relation-like Function-like set
B is Relation-like Function-like set
X +* B is Relation-like Function-like set
uB is set
(X +* B) " uB is set
X " uB is set
B " uB is set
(X " uB) \/ (B " uB) is set
x is set
proj1 (X +* B) is set
proj1 X is set
proj1 B is set
(proj1 X) \/ (proj1 B) is set
(X +* B) . x is set
X . x is set
B . x is set
X is Relation-like Function-like set
proj1 X is set
B is Relation-like Function-like set
proj1 B is set
X +* B is Relation-like Function-like set
uB is set
(X +* B) " uB is set
X " uB is set
B " uB is set
(X " uB) \/ (B " uB) is set
B is set
X is set
uB is Relation-like Function-like set
proj1 uB is set
X .--> uB is Relation-like {X} -defined Function-like one-to-one finite Function-yielding V65() set
{X} is non empty trivial finite 1 -element set
{X} --> uB is Relation-like {X} -defined {uB} -valued Function-like constant non empty total quasi_total finite Function-yielding V65() Element of bool [:{X},{uB}:]
{uB} is functional non empty trivial finite 1 -element set
[:{X},{uB}:] is Relation-like non empty finite set
bool [:{X},{uB}:] is non empty finite V37() set
commute (X .--> uB) is Relation-like Function-like Function-yielding V65() set
uncurry (X .--> uB) is Relation-like Function-like set
curry' (uncurry (X .--> uB)) is Relation-like Function-like set
(commute (X .--> uB)) . B is Relation-like Function-like set
uB . B is set
X .--> (uB . B) is Relation-like {X} -defined Function-like one-to-one finite set
{X} --> (uB . B) is Relation-like {X} -defined {(uB . B)} -valued Function-like constant non empty total quasi_total finite Element of bool [:{X},{(uB . B)}:]
{(uB . B)} is non empty trivial finite 1 -element set
[:{X},{(uB . B)}:] is Relation-like non empty finite set
bool [:{X},{(uB . B)}:] is non empty finite V37() set
dom (X .--> uB) is trivial finite Element of bool {X}
bool {X} is non empty finite V37() set
proj2 uB is set
Funcs ((proj1 uB),(proj2 uB)) is functional set
proj2 (X .--> uB) is finite set
Funcs ({X},(Funcs ((proj1 uB),(proj2 uB)))) is functional set
(X .--> uB) . X is Relation-like Function-like set
((commute (X .--> uB)) . B) . X is set
proj1 ((commute (X .--> uB)) . B) is set
X is set
B is Relation-like Function-like set
commute B is Relation-like Function-like Function-yielding V65() set
uncurry B is Relation-like Function-like set
curry' (uncurry B) is Relation-like Function-like set
proj1 (commute B) is set
proj1 B is set
proj1 (uncurry B) is set
proj2 (proj1 (uncurry B)) is set
uB is set
[uB,X] is V22() set
{uB,X} is finite set
{uB} is non empty trivial finite 1 -element set
{{uB,X},{uB}} is finite V37() set
x is set
V9 is set
[x,V9] is V22() set
{x,V9} is finite set
{x} is non empty trivial finite 1 -element set
{{x,V9},{x}} is finite V37() set
V is Relation-like Function-like set
B . x is set
proj1 V is set
x9 is set
r is Relation-like Function-like set
B . x9 is set
proj1 r is set
uB is set
x is Relation-like Function-like set
B . uB is set
proj1 x is set
[uB,X] is V22() set
{uB,X} is finite set
{uB} is non empty trivial finite 1 -element set
{{uB,X},{uB}} is finite V37() set
X is set
B is set
uB is Relation-like Function-like set
commute uB is Relation-like Function-like Function-yielding V65() set
uncurry uB is Relation-like Function-like set
curry' (uncurry uB) is Relation-like Function-like set
(commute uB) . B is Relation-like Function-like set
proj1 ((commute uB) . B) is set
proj1 uB is set
uB . X is set
proj1 (uncurry uB) is set
proj2 uB is set
union (proj2 uB) is set
proj1 (union (proj2 uB)) is set
[:(proj1 uB),(proj1 (union (proj2 uB))):] is Relation-like set
x is set
V is set
x9 is set
[V,x9] is V22() set
{V,x9} is finite set
{V} is non empty trivial finite 1 -element set
{{V,x9},{V}} is finite V37() set
V9 is Relation-like Function-like set
uB . V is set
proj1 V9 is set
uncurry' (commute uB) is Relation-like Function-like set
proj1 (commute uB) is set
[X,B] is V22() set
{X,B} is finite set
{X} is non empty trivial finite 1 -element set
{{X,B},{X}} is finite V37() set
x is set
V9 is set
[x,V9] is V22() set
{x,V9} is finite set
{x} is non empty trivial finite 1 -element set
{{x,V9},{x}} is finite V37() set
V is Relation-like Function-like set
uB . x is set
proj1 V is set
x9 is Relation-like Function-like set
proj1 x9 is set
x is Relation-like Function-like set
proj1 x is set
X is set
B is set
uB is Relation-like Function-like set
proj1 uB is set
x is Relation-like Function-like set
uB . X is set
proj1 x is set
commute uB is Relation-like Function-like Function-yielding V65() set
uncurry uB is Relation-like Function-like set
curry' (uncurry uB) is Relation-like Function-like set
(commute uB) . B is Relation-like Function-like set
((commute uB) . B) . X is set
x . B is set
[X,B] is V22() set
{X,B} is finite set
{X} is non empty trivial finite 1 -element set
{{X,B},{X}} is finite V37() set
proj1 (uncurry uB) is set
[X,B] `2 is set
[X,B] `1 is set
(uncurry uB) . (X,B) is set
(uncurry uB) . [X,B] is set
X is set
x is Relation-like Function-like set
B is Relation-like Function-like set
uB is Relation-like Function-like set
B \/ uB is Relation-like set
commute x is Relation-like Function-like Function-yielding V65() set
uncurry x is Relation-like Function-like set
curry' (uncurry x) is Relation-like Function-like set
(commute x) . X is Relation-like Function-like set
commute B is Relation-like Function-like Function-yielding V65() set
uncurry B is Relation-like Function-like set
curry' (uncurry B) is Relation-like Function-like set
(commute B) . X is Relation-like Function-like set
commute uB is Relation-like Function-like Function-yielding V65() set
uncurry uB is Relation-like Function-like set
curry' (uncurry uB) is Relation-like Function-like set
(commute uB) . X is Relation-like Function-like set
((commute B) . X) \/ ((commute uB) . X) is Relation-like set
V is set
V9 is set
[V,V9] is V22() set
{V,V9} is finite set
{V} is non empty trivial finite 1 -element set
{{V,V9},{V}} is finite V37() set
((commute x) . X) . V is set
proj1 ((commute x) . X) is set
proj1 x is set
x . V is set
x9 is Relation-like Function-like set
proj1 x9 is set
[V,x9] is V22() set
{V,x9} is finite set
{{V,x9},{V}} is finite V37() set
proj1 B is set
B . V is set
proj1 uB is set
uB . V is set
proj1 ((commute B) . X) is set
((commute B) . X) . V is set
x9 . X is set
proj1 ((commute uB) . X) is set
((commute uB) . X) . V is set
x9 is Relation-like Function-like set
proj1 x9 is set
x9 . X is set
[V,x9] is V22() set
{V,x9} is finite set
{{V,x9},{V}} is finite V37() set
x9 is Relation-like Function-like set
proj1 x9 is set
x9 . X is set
[V,x9] is V22() set
{V,x9} is finite set
{{V,x9},{V}} is finite V37() set
X is set
B is set
<*X,B*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
product <*X,B*> is set
[:X,B:] is Relation-like set
len (product <*X,B*>) is ordinal cardinal set
len X is ordinal cardinal set
len B is ordinal cardinal set
(len X) *` (len B) is ordinal cardinal set
uB is Relation-like Function-like set
proj1 uB is set
<*X,B*> . 2 is set
dom <*X,B*> is non empty finite 2 -element V169() V170() V171() V172() V173() V174() V267() V268() V269() V270() V271() Element of bool NAT
{1,2} is finite V37() V169() V170() V171() V172() V173() V174() V269() V270() V271() Element of bool REAL
<*X,B*> . 1 is set
proj2 uB is set
x is set
V is set
uB . x is set
uB . V is set
V9 is Relation-like Function-like set
proj1 V9 is set
x9 is Relation-like Function-like set
proj1 x9 is set
x9 . 1 is set
x9 . 2 is set
[(x9 . 1),(x9 . 2)] is V22() set
{(x9 . 1),(x9 . 2)} is finite set
{(x9 . 1)} is non empty trivial finite 1 -element set
{{(x9 . 1),(x9 . 2)},{(x9 . 1)}} is finite V37() set
V9 . 1 is set
V9 . 2 is set
[(V9 . 1),(V9 . 2)] is V22() set
{(V9 . 1),(V9 . 2)} is finite set
{(V9 . 1)} is non empty trivial finite 1 -element set
{{(V9 . 1),(V9 . 2)},{(V9 . 1)}} is finite V37() set
r is set
x9 . r is set
V9 . r is set
x is set
V is set
uB . V is set
V9 is Relation-like Function-like set
proj1 V9 is set
V9 . 1 is set
V9 . 2 is set
[(V9 . 1),(V9 . 2)] is V22() set
{(V9 . 1),(V9 . 2)} is finite set
{(V9 . 1)} is non empty trivial finite 1 -element set
{{(V9 . 1),(V9 . 2)},{(V9 . 1)}} is finite V37() set
x is set
x `1 is set
x `2 is set
<*(x `1),(x `2)*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*(x `1),(x `2)*> . 1 is set
<*(x `1),(x `2)*> . 2 is set
[(x `1),(x `2)] is V22() set
{(x `1),(x `2)} is finite set
{(x `1)} is non empty trivial finite 1 -element set
{{(x `1),(x `2)},{(x `1)}} is finite V37() set
uB . <*(x `1),(x `2)*> is set
len [:X,B:] is ordinal cardinal set
[:(len X),(len B):] is Relation-like set
len [:(len X),(len B):] is ordinal cardinal set
F<sub>3</sub>() is non empty set
F<sub>1</sub>() is non empty set
F<sub>2</sub>() is non empty set
[:F<sub>3</sub>(),F<sub>2</sub>():] is Relation-like non empty set
bool [:F<sub>3</sub>(),F<sub>2</sub>():] is non empty set
X is set
F<sub>4</sub>(X) is set
F<sub>5</sub>(X) is set
X is Relation-like F<sub>3</sub>() -defined F<sub>2</sub>() -valued Function-like non empty total quasi_total Element of bool [:F<sub>3</sub>(),F<sub>2</sub>():]
B is Element of F<sub>1</sub>()
X . B is set
F<sub>4</sub>(B) is set
F<sub>5</sub>(B) is set
F<sub>3</sub>() is non empty set
F<sub>1</sub>() is non empty set
F<sub>2</sub>() is non empty set
[:F<sub>3</sub>(),F<sub>2</sub>():] is Relation-like non empty set
bool [:F<sub>3</sub>(),F<sub>2</sub>():] is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is Element of F<sub>3</sub>() : ( P<sub>1</sub>[b<sub>1</sub>] or P<sub>2</sub>[b<sub>1</sub>] ) } is set
B is Relation-like Function-like set
proj1 B is set
proj2 B is set
uB is set
x is set
B . x is set
F<sub>6</sub>(x) is set
uB is Relation-like F<sub>3</sub>() -defined F<sub>2</sub>() -valued Function-like non empty total quasi_total Element of bool [:F<sub>3</sub>(),F<sub>2</sub>():]
x is Element of F<sub>1</sub>()
uB . x is set
F<sub>4</sub>(x) is set
F<sub>5</sub>(x) is set
F<sub>6</sub>(x) is set
B is non empty set
uB is set
F<sub>4</sub>(uB) is set
F<sub>5</sub>(uB) is set
x is Element of F<sub>3</sub>()
x is Element of F<sub>3</sub>()
[:B,F<sub>2</sub>():] is Relation-like non empty set
bool [:B,F<sub>2</sub>():] is non empty set
uB is Relation-like B -defined F<sub>2</sub>() -valued Function-like non empty total quasi_total Element of bool [:B,F<sub>2</sub>():]
x is set
uB . x is set
F<sub>6</sub>(x) is set
F<sub>5</sub>(x) is set
F<sub>4</sub>(x) is set
x is Relation-like F<sub>3</sub>() -defined F<sub>2</sub>() -valued Function-like non empty total quasi_total Element of bool [:F<sub>3</sub>(),F<sub>2</sub>():]
V is Element of F<sub>1</sub>()
x . V is set
F<sub>4</sub>(V) is set
F<sub>5</sub>(V) is set
F<sub>6</sub>(V) is set
uB . V is set
X is V28() real ext-real set
B is V28() real ext-real set
|[X,B]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|.|[X,B]|.| is V28() real ext-real non negative Element of REAL
|.|[X,B]|.| ^2 is V28() real ext-real Element of REAL
|.|[X,B]|.| * |.|[X,B]|.| is V28() real ext-real non negative set
X ^2 is V28() real ext-real set
X * X is V28() real ext-real set
B ^2 is V28() real ext-real set
B * B is V28() real ext-real set
(X ^2) + (B ^2) is V28() real ext-real set
|[X,B]| `2 is V28() real ext-real Element of REAL
|[X,B]| `1 is V28() real ext-real Element of REAL
X is TopSpace-like TopStruct
the carrier of X is set
bool the carrier of X is non empty set
B is non empty TopSpace-like TopStruct
the carrier of B is non empty set
uB is closed Element of bool the carrier of X
X | uB is strict TopSpace-like SubSpace of X
the carrier of (X | uB) is set
[: the carrier of (X | uB), the carrier of B:] is Relation-like set
bool [: the carrier of (X | uB), the carrier of B:] is non empty set
x is closed Element of bool the carrier of X
X | x is strict TopSpace-like SubSpace of X
the carrier of (X | x) is set
[: the carrier of (X | x), the carrier of B:] is Relation-like set
bool [: the carrier of (X | x), the carrier of B:] is non empty set
uB \/ x is closed Element of bool the carrier of X
X | (uB \/ x) is strict TopSpace-like SubSpace of X
the carrier of (X | (uB \/ x)) is set
[: the carrier of (X | (uB \/ x)), the carrier of B:] is Relation-like set
bool [: the carrier of (X | (uB \/ x)), the carrier of B:] is non empty set
V is Relation-like the carrier of (X | uB) -defined the carrier of B -valued Function-like total quasi_total continuous Element of bool [: the carrier of (X | uB), the carrier of B:]
V9 is Relation-like the carrier of (X | x) -defined the carrier of B -valued Function-like total quasi_total continuous Element of bool [: the carrier of (X | x), the carrier of B:]
V +* V9 is Relation-like Function-like set
dom V9 is Element of bool the carrier of (X | x)
bool the carrier of (X | x) is non empty set
dom V is Element of bool the carrier of (X | uB)
bool the carrier of (X | uB) is non empty set
proj2 (V +* V9) is set
rng V is Element of bool the carrier of B
bool the carrier of B is non empty set
rng V9 is Element of bool the carrier of B
(rng V) \/ (rng V9) is Element of bool the carrier of B
proj1 (V +* V9) is set
(dom V) \/ (dom V9) is set
x9 is Relation-like the carrier of (X | (uB \/ x)) -defined the carrier of B -valued Function-like total quasi_total Element of bool [: the carrier of (X | (uB \/ x)), the carrier of B:]
r is Element of bool the carrier of B
x9 " r is Element of bool the carrier of (X | (uB \/ x))
bool the carrier of (X | (uB \/ x)) is non empty set
[#] (X | (uB \/ x)) is non proper open closed dense Element of bool the carrier of (X | (uB \/ x))
V " r is Element of bool the carrier of (X | uB)
[#] (X | uB) is non proper open closed dense Element of bool the carrier of (X | uB)
f is Element of bool the carrier of X
f /\ ([#] (X | uB)) is Element of bool the carrier of (X | uB)
V9 " r is Element of bool the carrier of (X | x)
[#] (X | x) is non proper open closed dense Element of bool the carrier of (X | x)
f is Element of bool the carrier of X
f /\ ([#] (X | x)) is Element of bool the carrier of (X | x)
f /\ uB is Element of bool the carrier of X
f /\ x is Element of bool the carrier of X
(f /\ uB) \/ (f /\ x) is Element of bool the carrier of X
(V " r) \/ (V9 " r) is set
((V " r) \/ (V9 " r)) /\ (uB \/ x) is Element of bool the carrier of X
X is TopSpace-like TopStruct
the carrier of X is set
bool the carrier of X is non empty set
B is non empty TopSpace-like TopStruct
the carrier of B is non empty set
uB is closed Element of bool the carrier of X
x is closed Element of bool the carrier of X
X | uB is strict TopSpace-like SubSpace of X
the carrier of (X | uB) is set
[: the carrier of (X | uB), the carrier of B:] is Relation-like set
bool [: the carrier of (X | uB), the carrier of B:] is non empty set
X | x is strict TopSpace-like SubSpace of X
the carrier of (X | x) is set
[: the carrier of (X | x), the carrier of B:] is Relation-like set
bool [: the carrier of (X | x), the carrier of B:] is non empty set
uB \/ x is closed Element of bool the carrier of X
X | (uB \/ x) is strict TopSpace-like SubSpace of X
the carrier of (X | (uB \/ x)) is set
[: the carrier of (X | (uB \/ x)), the carrier of B:] is Relation-like set
bool [: the carrier of (X | (uB \/ x)), the carrier of B:] is non empty set
V is Relation-like the carrier of (X | uB) -defined the carrier of B -valued Function-like total quasi_total continuous Element of bool [: the carrier of (X | uB), the carrier of B:]
V9 is Relation-like the carrier of (X | x) -defined the carrier of B -valued Function-like total quasi_total continuous Element of bool [: the carrier of (X | x), the carrier of B:]
V +* V9 is Relation-like Function-like set
dom V9 is Element of bool the carrier of (X | x)
bool the carrier of (X | x) is non empty set
dom V is Element of bool the carrier of (X | uB)
bool the carrier of (X | uB) is non empty set
X is TopSpace-like TopStruct
the carrier of X is set
bool the carrier of X is non empty set
B is non empty TopSpace-like TopStruct
the carrier of B is non empty set
[: the carrier of X, the carrier of B:] is Relation-like set
bool [: the carrier of X, the carrier of B:] is non empty set
uB is open closed Element of bool the carrier of X
X | uB is strict TopSpace-like SubSpace of X
the carrier of (X | uB) is set
[: the carrier of (X | uB), the carrier of B:] is Relation-like set
bool [: the carrier of (X | uB), the carrier of B:] is non empty set
uB ` is open closed Element of bool the carrier of X
the carrier of X \ uB is set
X | (uB `) is strict TopSpace-like SubSpace of X
the carrier of (X | (uB `)) is set
[: the carrier of (X | (uB `)), the carrier of B:] is Relation-like set
bool [: the carrier of (X | (uB `)), the carrier of B:] is non empty set
x is Relation-like the carrier of (X | uB) -defined the carrier of B -valued Function-like total quasi_total continuous Element of bool [: the carrier of (X | uB), the carrier of B:]
V is Relation-like the carrier of (X | (uB `)) -defined the carrier of B -valued Function-like total quasi_total continuous Element of bool [: the carrier of (X | (uB `)), the carrier of B:]
x +* V is Relation-like Function-like set
uB \/ (uB `) is open closed Element of bool the carrier of X
[#] X is non proper open closed dense Element of bool the carrier of X
X | (uB \/ (uB `)) is strict TopSpace-like SubSpace of X
the topology of X is non empty Element of bool (bool the carrier of X)
bool (bool the carrier of X) is non empty set
TopStruct(# the carrier of X, the topology of X #) is strict TopSpace-like TopStruct
the carrier of (X | (uB \/ (uB `))) is set
[: the carrier of (X | (uB \/ (uB `))), the carrier of B:] is Relation-like set
bool [: the carrier of (X | (uB \/ (uB `))), the carrier of B:] is non empty set
the topology of B is non empty Element of bool (bool the carrier of B)
bool the carrier of B is non empty set
bool (bool the carrier of B) is non empty set
TopStruct(# the carrier of B, the topology of B #) is non empty strict TopSpace-like TopStruct
X is ordinal natural V28() real ext-real finite cardinal V99() rational V169() V170() V171() V172() V173() V174() V269() V270() V271() Element of NAT
TOP-REAL X is non empty TopSpace-like V116() V181() V182() V241() V242() V243() V244() V245() V246() V247() V253() L16()
the carrier of (TOP-REAL X) is non empty set
B is Relation-like NAT -defined Function-like finite X -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL X)
uB is non empty V28() real ext-real positive non negative set
Ball (B,uB) is Element of bool the carrier of (TOP-REAL X)
bool the carrier of (TOP-REAL X) is non empty set
B - B is Relation-like NAT -defined Function-like finite X -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL X)
K391(B,B) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
0. (TOP-REAL X) is Relation-like NAT -defined Function-like finite X -element V74( TOP-REAL X) FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL X)
the ZeroF of (TOP-REAL X) is Relation-like NAT -defined Function-like finite X -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL X)
|.(B - B).| is V28() real ext-real non negative Element of REAL
{ |[b₁,0]| where b₁ is V28() real ext-real Element of REAL : verum } is set
bool the carrier of (TOP-REAL 2) is non empty set
B is set
uB is V28() real ext-real Element of REAL
|[uB,0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
{ |[b₁,b₂]| where b₁, b₂ is V28() real ext-real Element of REAL : 0 <= b₂ } is set
B is set
uB is V28() real ext-real Element of REAL
x is V28() real ext-real Element of REAL
|[uB,x]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
() is Element of bool the carrier of (TOP-REAL 2)
() is Element of bool the carrier of (TOP-REAL 2)
X is set
B is set
<*X,B*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
uB is V28() real ext-real Element of REAL
|[uB,0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
<*X,B*> . 1 is set
uB is V28() real ext-real Element of REAL
|[uB,0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
uB is V28() real ext-real Element of REAL
|[uB,0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
<*X,B*> . 2 is set
uB is V28() real ext-real Element of REAL
|[uB,0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
<*X,0*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
X is V28() real ext-real set
B is V28() real ext-real set
|[X,B]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
uB is V28() real ext-real Element of REAL
|[uB,0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
uB is V28() real ext-real Element of REAL
|[uB,0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
len () is ordinal cardinal set
X is Relation-like Function-like set
proj1 X is set
proj2 X is set
B is set
uB is set
X . B is set
X . uB is set
V is V28() real ext-real Element of REAL
X . V is set
|[V,0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
x is V28() real ext-real Element of REAL
X . x is set
|[x,0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
B is set
uB is set
X . uB is set
x is V28() real ext-real Element of REAL
|[x,0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
B is set
uB is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
uB `1 is V28() real ext-real Element of REAL
uB `2 is V28() real ext-real Element of REAL
|[(uB `1),(uB `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
X . (uB `1) is set
X is set
B is set
<*X,B*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*X,B*> . 1 is set
uB is V28() real ext-real Element of REAL
x is V28() real ext-real Element of REAL
|[uB,x]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
<*X,B*> . 2 is set
V is V28() real ext-real Element of REAL
x is V28() real ext-real Element of REAL
uB is V28() real ext-real Element of REAL
|[uB,x]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
X is V28() real ext-real set
B is V28() real ext-real set
|[X,B]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
uB is V28() real ext-real Element of REAL
x is V28() real ext-real Element of REAL
|[uB,x]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
uB is V28() real ext-real Element of REAL
x is V28() real ext-real Element of REAL
|[uB,x]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
X is set
B is Element of ()
B `1 is V28() real ext-real Element of REAL
B `2 is V28() real ext-real Element of REAL
|[(B `1),(B `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
uB is V28() real ext-real set
X is V28() real ext-real set
B is V28() real ext-real set
|[X,B]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[X,B]|,uB) is Element of bool the carrier of (TOP-REAL 2)
x is V28() real ext-real non negative Element of REAL
x - uB is V28() real ext-real Element of REAL
V is non empty V28() real ext-real non positive negative Element of REAL
V / 2 is non empty V28() real ext-real non positive negative Element of REAL
|[X,(V / 2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[X,x]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[X,(V / 2)]| - |[X,x]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[X,(V / 2)]|,|[X,x]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
X - X is V28() real ext-real set
(V / 2) - x is non empty V28() real ext-real non positive negative Element of REAL
|[(X - X),((V / 2) - x)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|.(|[X,(V / 2)]| - |[X,x]|).| is V28() real ext-real non negative Element of REAL
|.((V / 2) - x).| is V28() real ext-real Element of REAL
x - (V / 2) is non empty V28() real ext-real positive non negative Element of REAL
|.(x - (V / 2)).| is V28() real ext-real Element of REAL
x9 is non empty V28() real ext-real positive non negative Element of REAL
x + x9 is non empty V28() real ext-real positive non negative Element of REAL
(x + x9) / 2 is non empty V28() real ext-real positive non negative Element of REAL
x9 + x9 is non empty V28() real ext-real positive non negative Element of REAL
(x9 + x9) / 2 is non empty V28() real ext-real positive non negative Element of REAL
Ball (|[X,x]|,x9) is Element of bool the carrier of (TOP-REAL 2)
x is set
V is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
V - |[X,B]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(V,|[X,B]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(V - |[X,B]|).| is V28() real ext-real non negative Element of REAL
V `1 is V28() real ext-real Element of REAL
(V `1) - X is V28() real ext-real Element of REAL
V `2 is V28() real ext-real Element of REAL
(V `2) - B is V28() real ext-real Element of REAL
|[((V `1) - X),((V `2) - B)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[((V `1) - X),((V `2) - B)]| `1 is V28() real ext-real Element of REAL
|[((V `1) - X),((V `2) - B)]| `2 is V28() real ext-real Element of REAL
|[(V `1),(V `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
((V `1) - X) ^2 is V28() real ext-real Element of REAL
((V `1) - X) * ((V `1) - X) is V28() real ext-real set
((V `2) - B) ^2 is V28() real ext-real Element of REAL
((V `2) - B) * ((V `2) - B) is V28() real ext-real set
(((V `1) - X) ^2) + (((V `2) - B) ^2) is V28() real ext-real Element of REAL
sqrt ((((V `1) - X) ^2) + (((V `2) - B) ^2)) is V28() real ext-real Element of REAL
abs ((V `2) - B) is V28() real ext-real Element of REAL
B - (V `2) is V28() real ext-real Element of REAL
abs (B - (V `2)) is V28() real ext-real Element of REAL
uB is V28() real ext-real set
B is V28() real ext-real set
X is V28() real ext-real set
|[X,B]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[X,B]|,uB) is Element of bool the carrier of (TOP-REAL 2)
|[X,0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[X,0]| - |[X,B]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[X,0]|,|[X,B]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
X - X is V28() real ext-real set
0 - B is V28() real ext-real Element of REAL
|[(X - X),(0 - B)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|.(|[X,0]| - |[X,B]|).| is V28() real ext-real non negative Element of REAL
|.(0 - B).| is V28() real ext-real Element of REAL
B - 0 is V28() real ext-real Element of REAL
|.(B - 0).| is V28() real ext-real Element of REAL
x is set
V is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
V `1 is V28() real ext-real Element of REAL
V `2 is V28() real ext-real Element of REAL
|[(V `1),(V `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
V - |[X,B]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(V,|[X,B]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
(V `1) - X is V28() real ext-real Element of REAL
|[((V `1) - X),(0 - B)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
(V - |[X,B]|) `2 is V28() real ext-real Element of REAL
(V - |[X,B]|) `1 is V28() real ext-real Element of REAL
|.(V - |[X,B]|).| is V28() real ext-real non negative Element of REAL
((V `1) - X) ^2 is V28() real ext-real Element of REAL
((V `1) - X) * ((V `1) - X) is V28() real ext-real set
(0 - B) ^2 is V28() real ext-real Element of REAL
(0 - B) * (0 - B) is V28() real ext-real set
(((V `1) - X) ^2) + ((0 - B) ^2) is V28() real ext-real Element of REAL
sqrt ((((V `1) - X) ^2) + ((0 - B) ^2)) is V28() real ext-real Element of REAL
abs (0 - B) is V28() real ext-real Element of REAL
abs (B - 0) is V28() real ext-real Element of REAL
abs B is V28() real ext-real Element of REAL
X is ordinal natural V28() real ext-real finite cardinal V99() rational V169() V170() V171() V172() V173() V174() V269() V270() V271() Element of NAT
TOP-REAL X is non empty TopSpace-like V116() V181() V182() V241() V242() V243() V244() V245() V246() V247() V253() L16()
the carrier of (TOP-REAL X) is non empty set
B is Relation-like NAT -defined Function-like finite X -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL X)
uB is Relation-like NAT -defined Function-like finite X -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL X)
B - uB is Relation-like NAT -defined Function-like finite X -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL X)
K391(B,uB) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(B - uB).| is V28() real ext-real non negative Element of REAL
x is non empty V28() real ext-real positive non negative set
V is non empty V28() real ext-real positive non negative set
x - V is V28() real ext-real set
Ball (uB,V) is Element of bool the carrier of (TOP-REAL X)
bool the carrier of (TOP-REAL X) is non empty set
Ball (B,x) is Element of bool the carrier of (TOP-REAL X)
uB - B is Relation-like NAT -defined Function-like finite X -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL X)
K391(uB,B) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(uB - B).| is V28() real ext-real non negative Element of REAL
V9 is set
x9 is Relation-like NAT -defined Function-like finite X -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL X)
x9 - uB is Relation-like NAT -defined Function-like finite X -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL X)
K391(x9,uB) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(x9 - uB).| is V28() real ext-real non negative Element of REAL
V + (x - V) is V28() real ext-real set
|.(x9 - uB).| + |.(uB - B).| is V28() real ext-real non negative Element of REAL
x9 - B is Relation-like NAT -defined Function-like finite X -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL X)
K391(x9,B) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(x9 - B).| is V28() real ext-real non negative Element of REAL
X is V28() real ext-real set
B is non empty V28() real ext-real positive non negative set
uB is non empty V28() real ext-real positive non negative set
|[X,B]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[X,B]|,B) is Element of bool the carrier of (TOP-REAL 2)
|[X,uB]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[X,uB]|,uB) is Element of bool the carrier of (TOP-REAL 2)
B - uB is V28() real ext-real set
uB - B is V28() real ext-real set
x is set
V is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
V - |[X,B]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(V,|[X,B]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(V - |[X,B]|).| is V28() real ext-real non negative Element of REAL
|[X,B]| - |[X,uB]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[X,B]|,|[X,uB]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
X - X is V28() real ext-real set
|[(X - X),(B - uB)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|.(|[X,B]| - |[X,uB]|).| is V28() real ext-real non negative Element of REAL
|.(B - uB).| is V28() real ext-real Element of REAL
|.(uB - B).| is V28() real ext-real Element of REAL
B + (uB - B) is V28() real ext-real set
|.(V - |[X,B]|).| + |.(|[X,B]| - |[X,uB]|).| is V28() real ext-real non negative Element of REAL
V - |[X,uB]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(V,|[X,uB]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(V - |[X,uB]|).| is V28() real ext-real non negative Element of REAL
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
B is non empty TopSpace-like TopStruct
the carrier of B is non empty set
the topology of X is non empty Element of bool (bool the carrier of X)
bool the carrier of X is non empty set
bool (bool the carrier of X) is non empty set
TopStruct(# the carrier of X, the topology of X #) is non empty strict TopSpace-like TopStruct
the topology of B is non empty Element of bool (bool the carrier of B)
bool the carrier of B is non empty set
bool (bool the carrier of B) is non empty set
TopStruct(# the carrier of B, the topology of B #) is non empty strict TopSpace-like TopStruct
uB is Relation-like the carrier of X -defined Function-like total Neighborhood_System of X
x is Relation-like the carrier of B -defined Function-like total Neighborhood_System of B
dom uB is Element of bool the carrier of X
dom x is Element of bool the carrier of B
Union x is open quasi_basis Element of bool (bool the carrier of B)
UniCl (Union x) is Element of bool (bool the carrier of B)
Union uB is open quasi_basis Element of bool (bool the carrier of X)
UniCl (Union uB) is Element of bool (bool the carrier of X)
{ ((Ball (|[(a<sub>1</sub> `1),b₁]|,b₁)) \/ {a₁}) where b₁ is V28() real ext-real Element of REAL : not b₁ <= 0 } is set
{ ((Ball (a₁,b₁)) /\ ()) where b₁ is V28() real ext-real Element of REAL : not b₁ <= 0 } is set
B is Relation-like () -defined Function-like total set
uB is set
B . uB is set
the non empty V28() real ext-real positive non negative Element of REAL is non empty V28() real ext-real positive non negative Element of REAL
x is Element of ()
x `1 is V28() real ext-real Element of REAL
|[(x `1), the non empty V28() real ext-real positive non negative Element of REAL ]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
B . x is set
{x} is non empty trivial finite 1 -element Element of bool the carrier of (TOP-REAL 2)
{ ((Ball (|[(x `1),b<sub>1</sub>]|,b<sub>1</sub>)) \/ {x}) where b<sub>1</sub> is V28() real ext-real Element of REAL : not b<sub>1</sub> <= 0 } is set
Ball (|[(x `1), the non empty V28() real ext-real positive non negative Element of REAL ]|, the non empty V28() real ext-real positive non negative Element of REAL ) is Element of bool the carrier of (TOP-REAL 2)
{x} is non empty trivial finite 1 -element Element of bool ()
bool () is non empty set
(Ball (|[(x `1), the non empty V28() real ext-real positive non negative Element of REAL ]|, the non empty V28() real ext-real positive non negative Element of REAL )) \/ {x} is set
the non empty V28() real ext-real positive non negative Element of REAL is non empty V28() real ext-real positive non negative Element of REAL
x is Element of ()
B . x is set
{ ((Ball (x,b<sub>1</sub>)) /\ ()) where b<sub>1</sub> is V28() real ext-real Element of REAL : not b<sub>1</sub> <= 0 } is set
Ball (x, the non empty V28() real ext-real positive non negative Element of REAL ) is Element of bool the carrier of (TOP-REAL 2)
(Ball (x, the non empty V28() real ext-real positive non negative Element of REAL )) /\ () is Element of bool the carrier of (TOP-REAL 2)
uB is Relation-like non-empty () -defined Function-like total set
proj2 uB is with_non-empty_elements set
bool () is non empty Element of bool (bool ())
bool () is non empty set
bool (bool ()) is non empty set
bool (bool ()) is non empty Element of bool (bool (bool ()))
bool (bool ()) is non empty set
bool (bool (bool ())) is non empty set
x is set
dom uB is Element of bool ()
V is set
uB . V is set
x9 is set
V9 is Element of ()
V9 `1 is V28() real ext-real Element of REAL
{V9} is non empty trivial finite 1 -element Element of bool the carrier of (TOP-REAL 2)
{ ((Ball (|[(V9 `1),b<sub>1</sub>]|,b<sub>1</sub>)) \/ {V9}) where b<sub>1</sub> is V28() real ext-real Element of REAL : not b<sub>1</sub> <= 0 } is set
{V9} is non empty trivial finite 1 -element Element of bool ()
r is V28() real ext-real Element of REAL
|[(V9 `1),r]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[(V9 `1),r]|,r) is Element of bool the carrier of (TOP-REAL 2)
(Ball (|[(V9 `1),r]|,r)) \/ {V9} is set
V9 is Element of ()
{ ((Ball (V9,b₁)) /\ ()) where b₁ is V28() real ext-real Element of REAL : not b₁ <= 0 } is set
r is V28() real ext-real Element of REAL
Ball (V9,r) is Element of bool the carrier of (TOP-REAL 2)
(Ball (V9,r)) /\ () is Element of bool the carrier of (TOP-REAL 2)
V9 is Element of ()
V9 `1 is V28() real ext-real Element of REAL
{V9} is non empty trivial finite 1 -element Element of bool the carrier of (TOP-REAL 2)
{ ((Ball (|[(V9 `1),b₁]|,b₁)) \/ {V9}) where b₁ is V28() real ext-real Element of REAL : not b₁ <= 0 } is set
{ ((Ball (V9,b₁)) /\ ()) where b₁ is V28() real ext-real Element of REAL : not b₁ <= 0 } is set
x is set
V9 is set
V is set
uB . V is set
uB . x is set
x9 is Element of ()
uB . x9 is set
x9 `1 is V28() real ext-real Element of REAL
{x9} is non empty trivial finite 1 -element Element of bool the carrier of (TOP-REAL 2)
{ ((Ball (|[(x9 `1),b₁]|,b₁)) \/ {x9}) where b₁ is V28() real ext-real Element of REAL : not b₁ <= 0 } is set
{x9} is non empty trivial finite 1 -element Element of bool ()
r is V28() real ext-real Element of REAL
|[(x9 `1),r]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[(x9 `1),r]|,r) is Element of bool the carrier of (TOP-REAL 2)
(Ball (|[(x9 `1),r]|,r)) \/ {x9} is set
f is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
f - |[(x9 `1),r]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(f,|[(x9 `1),r]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(f - |[(x9 `1),r]|).| is V28() real ext-real non negative Element of REAL
r - |.(f - |[(x9 `1),r]|).| is V28() real ext-real Element of REAL
Ball (f,(r - |.(f - |[(x9 `1),r]|).|)) is Element of bool the carrier of (TOP-REAL 2)
(Ball (f,(r - |.(f - |[(x9 `1),r]|).|))) /\ () is Element of bool the carrier of (TOP-REAL 2)
c is non empty V28() real ext-real positive non negative Element of REAL
uB . f is set
{ ((Ball (f,b<sub>1</sub>)) /\ ()) where b<sub>1</sub> is V28() real ext-real Element of REAL : not b<sub>1</sub> <= 0 } is set
u is Element of bool the carrier of (TOP-REAL 2)
b is non empty V28() real ext-real positive non negative Element of REAL
b - c is V28() real ext-real Element of REAL
|[(x9 `1),r]| - f is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[(x9 `1),r]|,f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[(x9 `1),r]| - f).| is V28() real ext-real non negative Element of REAL
Ball (f,c) is Element of bool the carrier of (TOP-REAL 2)
Ball (|[(x9 `1),r]|,b) is Element of bool the carrier of (TOP-REAL 2)
f is Element of bool the carrier of (TOP-REAL 2)
x9 is Element of ()
uB . x9 is set
{ ((Ball (x9,b<sub>1</sub>)) /\ ()) where b<sub>1</sub> is V28() real ext-real Element of REAL : not b<sub>1</sub> <= 0 } is set
r is V28() real ext-real Element of REAL
Ball (x9,r) is Element of bool the carrier of (TOP-REAL 2)
(Ball (x9,r)) /\ () is Element of bool the carrier of (TOP-REAL 2)
f is Element of ()
f - x9 is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(f,x9) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(f - x9).| is V28() real ext-real non negative Element of REAL
r - |.(f - x9).| is V28() real ext-real Element of REAL
f `1 is V28() real ext-real Element of REAL
f `2 is V28() real ext-real Element of REAL
|[(f `1),(f `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
b is non empty V28() real ext-real positive non negative Element of REAL
b / 2 is non empty V28() real ext-real positive non negative Element of REAL
|[(f `1),(b / 2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[(f `1),(b / 2)]| - f is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[(f `1),(b / 2)]|,f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
(f `1) - (f `1) is V28() real ext-real Element of REAL
(b / 2) - 0 is V28() real ext-real Element of REAL
|[((f `1) - (f `1)),((b / 2) - 0)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|.(|[(f `1),(b / 2)]| - f).| is V28() real ext-real non negative Element of REAL
abs (b / 2) is V28() real ext-real Element of REAL
|[(f `1),(b / 2)]| - x9 is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[(f `1),(b / 2)]|,x9) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[(f `1),(b / 2)]| - x9).| is V28() real ext-real non negative Element of REAL
(b / 2) + |.(f - x9).| is non empty V28() real ext-real positive non negative Element of REAL
x9 - |[(f `1),(b / 2)]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(x9,|[(f `1),(b / 2)]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(x9 - |[(f `1),(b / 2)]|).| is V28() real ext-real non negative Element of REAL
r - (b / 2) is V28() real ext-real Element of REAL
Ball (|[(f `1),(b / 2)]|,(b / 2)) is Element of bool the carrier of (TOP-REAL 2)
u is non empty V28() real ext-real positive non negative Element of REAL
Ball (x9,u) is Element of bool the carrier of (TOP-REAL 2)
{f} is non empty trivial finite 1 -element Element of bool ()
(Ball (|[(f `1),(b / 2)]|,(b / 2))) \/ {f} is set
uB . f is set
{f} is non empty trivial finite 1 -element Element of bool the carrier of (TOP-REAL 2)
{ ((Ball (|[(f `1),b₁]|,b₁)) \/ {f}) where b₁ is V28() real ext-real Element of REAL : not b₁ <= 0 } is set
b is non empty V28() real ext-real positive non negative Element of REAL
Ball (f,b) is Element of bool the carrier of (TOP-REAL 2)
(Ball (f,b)) /\ () is Element of bool the carrier of (TOP-REAL 2)
c is Element of bool the carrier of (TOP-REAL 2)
uB . f is set
{ ((Ball (f,b₁)) /\ ()) where b₁ is V28() real ext-real Element of REAL : not b₁ <= 0 } is set
x9 - f is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(x9,f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(x9 - f).| is V28() real ext-real non negative Element of REAL
u is non empty V28() real ext-real positive non negative Element of REAL
u - b is V28() real ext-real Element of REAL
x9 is Element of ()
x is set
V is set
uB . x is set
V9 is set
V /\ V9 is set
x9 is Element of ()
x9 `1 is V28() real ext-real Element of REAL
{x9} is non empty trivial finite 1 -element Element of bool the carrier of (TOP-REAL 2)
{ ((Ball (|[(x9 `1),b₁]|,b₁)) \/ {x9}) where b₁ is V28() real ext-real Element of REAL : not b₁ <= 0 } is set
{x9} is non empty trivial finite 1 -element Element of bool ()
r is V28() real ext-real Element of REAL
|[(x9 `1),r]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[(x9 `1),r]|,r) is Element of bool the carrier of (TOP-REAL 2)
(Ball (|[(x9 `1),r]|,r)) \/ {x9} is set
f is V28() real ext-real Element of REAL
|[(x9 `1),f]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[(x9 `1),f]|,f) is Element of bool the carrier of (TOP-REAL 2)
(Ball (|[(x9 `1),f]|,f)) \/ {x9} is set
f is set
x9 is Element of ()
{ ((Ball (x9,b₁)) /\ ()) where b₁ is V28() real ext-real Element of REAL : not b₁ <= 0 } is set
r is V28() real ext-real Element of REAL
Ball (x9,r) is Element of bool the carrier of (TOP-REAL 2)
(Ball (x9,r)) /\ () is Element of bool the carrier of (TOP-REAL 2)
f is V28() real ext-real Element of REAL
Ball (x9,f) is Element of bool the carrier of (TOP-REAL 2)
(Ball (x9,f)) /\ () is Element of bool the carrier of (TOP-REAL 2)
f is set
x9 is Element of ()
x is set
V is set
uB . x is set
V9 is Element of ()
V9 `1 is V28() real ext-real Element of REAL
{V9} is non empty trivial finite 1 -element Element of bool the carrier of (TOP-REAL 2)
{ ((Ball (|[(V9 `1),b₁]|,b₁)) \/ {V9}) where b₁ is V28() real ext-real Element of REAL : not b₁ <= 0 } is set
{V9} is non empty trivial finite 1 -element Element of bool ()
x9 is V28() real ext-real Element of REAL
|[(V9 `1),x9]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[(V9 `1),x9]|,x9) is Element of bool the carrier of (TOP-REAL 2)
(Ball (|[(V9 `1),x9]|,x9)) \/ {V9} is set
V9 is Element of ()
{ ((Ball (V9,b<sub>1</sub>)) /\ ()) where b<sub>1</sub> is V28() real ext-real Element of REAL : not b<sub>1</sub> <= 0 } is set
x9 is V28() real ext-real Element of REAL
Ball (V9,x9) is Element of bool the carrier of (TOP-REAL 2)
(Ball (V9,x9)) /\ () is Element of bool the carrier of (TOP-REAL 2)
V9 is Element of ()
V9 `1 is V28() real ext-real Element of REAL
{V9} is non empty trivial finite 1 -element Element of bool the carrier of (TOP-REAL 2)
{ ((Ball (|[(V9 `1),b<sub>1</sub>]|,b<sub>1</sub>)) \/ {V9}) where b<sub>1</sub> is V28() real ext-real Element of REAL : not b<sub>1</sub> <= 0 } is set
{ ((Ball (V9,b<sub>1</sub>)) /\ ()) where b<sub>1</sub> is V28() real ext-real Element of REAL : not b<sub>1</sub> <= 0 } is set
Union uB is set
x is Element of bool (bool ())
UniCl x is Element of bool (bool ())
TopStruct(# (),(UniCl x) #) is non empty strict TopStruct
V9 is non empty strict TopSpace-like TopStruct
the carrier of V9 is non empty set
x9 is Relation-like the carrier of V9 -defined Function-like total Neighborhood_System of V9
r is V28() real ext-real Element of REAL
|[r,0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
{|[r,0]|} is functional non empty trivial finite V37() 1 -element Element of bool the carrier of (TOP-REAL 2)
|[r,0]| `1 is V28() real ext-real Element of REAL
f is V28() real ext-real Element of REAL
|[r,f]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[r,f]|,f) is Element of bool the carrier of (TOP-REAL 2)
(Ball (|[r,f]|,f)) \/ {|[r,0]|} is Element of bool the carrier of (TOP-REAL 2)
|[(|[r,0]| `1),f]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[(|[r,0]| `1),f]|,f) is Element of bool the carrier of (TOP-REAL 2)
(Ball (|[(|[r,0]| `1),f]|,f)) \/ {|[r,0]|} is Element of bool the carrier of (TOP-REAL 2)
{ H<sub>3</sub>(b<sub>1</sub>) where b<sub>1</sub> is V28() real ext-real Element of REAL : S<sub>2</sub>[b<sub>1</sub>] } is set
{ H<sub>4</sub>(b<sub>1</sub>) where b<sub>1</sub> is V28() real ext-real Element of REAL : S<sub>2</sub>[b<sub>1</sub>] } is set
x9 . |[r,0]| is set
{ ((Ball (|[r,b<sub>1</sub>]|,b<sub>1</sub>)) \/ {|[r,0]|}) where b<sub>1</sub> is V28() real ext-real Element of REAL : not b<sub>1</sub> <= 0 } is set
r is V28() real ext-real Element of REAL
f is V28() real ext-real Element of REAL
|[r,f]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
x9 . |[r,f]| is set
{ ((Ball (|[r,f]|,b<sub>1</sub>)) /\ ()) where b<sub>1</sub> is V28() real ext-real Element of REAL : not b<sub>1</sub> <= 0 } is set
X is non empty strict TopSpace-like TopStruct
the carrier of X is non empty set
B is non empty strict TopSpace-like TopStruct
the carrier of B is non empty set
uB is Relation-like the carrier of B -defined Function-like total Neighborhood_System of B
uB is Relation-like the carrier of B -defined Function-like total Neighborhood_System of B
x is Relation-like the carrier of X -defined Function-like total Neighborhood_System of X
x is Relation-like the carrier of X -defined Function-like total Neighborhood_System of X
V is set
V9 is Element of ()
V9 `1 is V28() real ext-real Element of REAL
V9 `2 is V28() real ext-real Element of REAL
|[(V9 `1),(V9 `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
x . V9 is set
|[(V9 `1),0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
{|[(V9 `1),0]|} is functional non empty trivial finite V37() 1 -element Element of bool the carrier of (TOP-REAL 2)
{ ((Ball (|[(V9 `1),b₁]|,b₁)) \/ {|[(V9 `1),0]|}) where b<sub>1</sub> is V28() real ext-real Element of REAL : not b<sub>1</sub> <= 0 } is set
uB . V9 is set
{ ((Ball (|[(V9 `1),(V9 `2)]|,b<sub>1</sub>)) /\ ()) where b<sub>1</sub> is V28() real ext-real Element of REAL : not b<sub>1</sub> <= 0 } is set
x . V is set
uB . V is set
() is non empty strict TopSpace-like TopStruct
() \ () is Element of bool the carrier of (TOP-REAL 2)
the carrier of () is non empty set
bool the carrier of () is non empty set
X is Relation-like the carrier of () -defined Function-like total Neighborhood_System of ()
uB is Element of the carrier of ()
B is Element of bool the carrier of ()
x is V28() real ext-real Element of REAL
V is V28() real ext-real Element of REAL
|[x,V]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[x,V]|,V) is Element of bool the carrier of (TOP-REAL 2)
(Ball (|[x,V]|,V)) /\ () is Element of bool the carrier of (TOP-REAL 2)
x9 is Element of bool the carrier of ()
{ ((Ball (|[x,V]|,b<sub>1</sub>)) /\ ()) where b<sub>1</sub> is V28() real ext-real Element of REAL : not b<sub>1</sub> <= 0 } is set
X . uB is open V235((),uB) Element of bool (bool the carrier of ())
bool (bool the carrier of ()) is non empty set
dom X is Element of bool the carrier of ()
r is Element of bool the carrier of ()
Union X is open quasi_basis Element of bool (bool the carrier of ())
X is Element of bool the carrier of ()
X ` is Element of bool the carrier of ()
the carrier of () \ X is set
B is open Element of bool the carrier of ()
B ` is closed Element of bool the carrier of ()
the carrier of () \ B is set
X is V28() real ext-real set
|[X,0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
{|[X,0]|} is functional non empty trivial finite V37() 1 -element Element of bool the carrier of (TOP-REAL 2)
B is non empty V28() real ext-real positive non negative set
|[X,B]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[X,B]|,B) is Element of bool the carrier of (TOP-REAL 2)
(Ball (|[X,B]|,B)) \/ {|[X,0]|} is Element of bool the carrier of (TOP-REAL 2)
x is Relation-like the carrier of () -defined Function-like total Neighborhood_System of ()
x . |[X,0]| is set
{ ((Ball (|[X,b<sub>1</sub>]|,b<sub>1</sub>)) \/ {|[X,0]|}) where b<sub>1</sub> is V28() real ext-real Element of REAL : not b<sub>1</sub> <= 0 } is set
uB is Element of the carrier of ()
x . uB is open V235((),uB) Element of bool (bool the carrier of ())
bool (bool the carrier of ()) is non empty set
X is V28() real ext-real set
B is non empty V28() real ext-real positive non negative set
|[X,B]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
uB is non empty V28() real ext-real positive non negative set
Ball (|[X,B]|,uB) is Element of bool the carrier of (TOP-REAL 2)
(Ball (|[X,B]|,uB)) /\ () is Element of bool the carrier of (TOP-REAL 2)
V is Relation-like the carrier of () -defined Function-like total Neighborhood_System of ()
V . |[X,B]| is set
{ ((Ball (|[X,B]|,b<sub>1</sub>)) /\ ()) where b<sub>1</sub> is V28() real ext-real Element of REAL : not b<sub>1</sub> <= 0 } is set
x is Element of the carrier of ()
V . x is open V235((),x) Element of bool (bool the carrier of ())
bool (bool the carrier of ()) is non empty set
B is V28() real ext-real set
X is V28() real ext-real set
|[X,B]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
uB is non empty V28() real ext-real positive non negative set
Ball (|[X,B]|,uB) is Element of bool the carrier of (TOP-REAL 2)
x is set
V is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
V `2 is V28() real ext-real Element of REAL
B - (V `2) is V28() real ext-real Element of REAL
abs (B - (V `2)) is V28() real ext-real Element of REAL
V `1 is V28() real ext-real Element of REAL
|[(V `1),(V `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
V - |[X,B]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(V,|[X,B]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
(V `1) - X is V28() real ext-real Element of REAL
(V `2) - B is V28() real ext-real Element of REAL
|[((V `1) - X),((V `2) - B)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
(V - |[X,B]|) `2 is V28() real ext-real Element of REAL
(V - |[X,B]|) `1 is V28() real ext-real Element of REAL
|.(V - |[X,B]|).| is V28() real ext-real non negative Element of REAL
((V `1) - X) ^2 is V28() real ext-real Element of REAL
((V `1) - X) * ((V `1) - X) is V28() real ext-real set
((V `2) - B) ^2 is V28() real ext-real Element of REAL
((V `2) - B) * ((V `2) - B) is V28() real ext-real set
(((V `1) - X) ^2) + (((V `2) - B) ^2) is V28() real ext-real Element of REAL
sqrt ((((V `1) - X) ^2) + (((V `2) - B) ^2)) is V28() real ext-real Element of REAL
abs ((V `2) - B) is V28() real ext-real Element of REAL
(Ball (|[X,B]|,uB)) /\ () is Element of bool the carrier of (TOP-REAL 2)
X is Element of the carrier of ()
B is non empty V28() real ext-real positive non negative set
uB is Relation-like the carrier of () -defined Function-like total Neighborhood_System of ()
x is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
x `1 is V28() real ext-real Element of REAL
x `2 is V28() real ext-real Element of REAL
|[(x `1),(x `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
uB . X is open V235((),X) Element of bool (bool the carrier of ())
bool (bool the carrier of ()) is non empty set
|[(x `1),0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
{|[(x `1),0]|} is functional non empty trivial finite V37() 1 -element Element of bool the carrier of (TOP-REAL 2)
{ ((Ball (|[(x `1),b<sub>1</sub>]|,b<sub>1</sub>)) \/ {|[(x `1),0]|}) where b₁ is V28() real ext-real Element of REAL : not b₁ <= 0 } is set
B / 2 is non empty V28() real ext-real positive non negative Element of REAL
|[(x `1),(B / 2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[(x `1),(B / 2)]|,(B / 2)) is Element of bool the carrier of (TOP-REAL 2)
(Ball (|[(x `1),(B / 2)]|,(B / 2))) \/ {|[(x `1),0]|} is Element of bool the carrier of (TOP-REAL 2)
the topology of () is non empty Element of bool (bool the carrier of ())
{X} is non empty trivial finite 1 -element Element of bool the carrier of ()
(Ball (|[(x `1),(B / 2)]|,(B / 2))) \/ {X} is set
V9 is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
V is open Element of bool the carrier of ()
(B / 2) + (B / 2) is non empty V28() real ext-real positive non negative Element of REAL
Ball (V9,(B / 2)) is Element of bool the carrier of (TOP-REAL 2)
x9 is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
x9 - V9 is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(x9,V9) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(x9 - V9).| is V28() real ext-real non negative Element of REAL
x - V9 is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(x,V9) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
(x `1) - (x `1) is V28() real ext-real Element of REAL
0 - (B / 2) is V28() real ext-real Element of REAL
|[((x `1) - (x `1)),(0 - (B / 2))]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|.(x - V9).| is V28() real ext-real non negative Element of REAL
|.(0 - (B / 2)).| is V28() real ext-real Element of REAL
(B / 2) - 0 is V28() real ext-real Element of REAL
|.((B / 2) - 0).| is V28() real ext-real Element of REAL
uB . X is open V235((),X) Element of bool (bool the carrier of ())
bool (bool the carrier of ()) is non empty set
{ ((Ball (|[(x `1),(x `2)]|,b<sub>1</sub>)) /\ ()) where b<sub>1</sub> is V28() real ext-real Element of REAL : not b<sub>1</sub> <= 0 } is set
B / 2 is non empty V28() real ext-real positive non negative Element of REAL
Ball (x,(B / 2)) is Element of bool the carrier of (TOP-REAL 2)
(Ball (x,(B / 2))) /\ () is Element of bool the carrier of (TOP-REAL 2)
the topology of () is non empty Element of bool (bool the carrier of ())
V9 is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
V is open Element of bool the carrier of ()
Ball (V9,(B / 2)) is Element of bool the carrier of (TOP-REAL 2)
x9 is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
x9 - V9 is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(x9,V9) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(x9 - V9).| is V28() real ext-real non negative Element of REAL
(B / 2) + (B / 2) is non empty V28() real ext-real positive non negative Element of REAL
X is V28() real ext-real set
B is V28() real ext-real set
|[X,B]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
uB is non empty V28() real ext-real positive non negative set
Ball (|[X,B]|,uB) is Element of bool the carrier of (TOP-REAL 2)
uB / 2 is non empty V28() real ext-real positive non negative Element of REAL
X + (uB / 2) is V28() real ext-real Element of REAL
x is V28() real ext-real rational set
x - X is V28() real ext-real set
B + (uB / 2) is V28() real ext-real Element of REAL
V is V28() real ext-real rational set
V - B is V28() real ext-real set
|[x,V]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[X,V]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[x,V]| - |[X,V]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[x,V]|,|[X,V]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
V - V is V28() real ext-real rational set
|[(x - X),(V - V)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|.(|[x,V]| - |[X,V]|).| is V28() real ext-real non negative Element of REAL
abs (x - X) is V28() real ext-real Element of REAL
(X + (uB / 2)) - X is V28() real ext-real Element of REAL
|[X,V]| - |[X,B]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[X,V]|,|[X,B]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
X - X is V28() real ext-real set
|[(X - X),(V - B)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[x,V]| - |[X,B]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[x,V]|,|[X,B]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[x,V]| - |[X,B]|).| is V28() real ext-real non negative Element of REAL
|.(|[X,V]| - |[X,B]|).| is V28() real ext-real non negative Element of REAL
|.(|[x,V]| - |[X,V]|).| + |.(|[X,V]| - |[X,B]|).| is V28() real ext-real non negative Element of REAL
|.(V - B).| is V28() real ext-real Element of REAL
(B + (uB / 2)) - B is V28() real ext-real Element of REAL
((X + (uB / 2)) - X) + ((B + (uB / 2)) - B) is V28() real ext-real Element of REAL
<*RAT,RAT*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
product <*RAT,RAT*> is set
(() \ ()) /\ (product <*RAT,RAT*>) is Element of bool the carrier of (TOP-REAL 2)
[#] () is non empty non proper open closed dense non boundary Element of bool the carrier of ()
X is Element of bool the carrier of ()
B is set
{B} is non empty trivial finite 1 -element set
X \ {B} is Element of bool the carrier of ()
Cl (X \ {B}) is closed Element of bool the carrier of ()
uB is set
x is Element of the carrier of ()
V9 is Relation-like the carrier of () -defined Function-like total Neighborhood_System of ()
V is Element of ()
V `1 is V28() real ext-real Element of REAL
V `2 is V28() real ext-real Element of REAL
|[(V `1),(V `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
V9 . x is open V235((),x) Element of bool (bool the carrier of ())
bool (bool the carrier of ()) is non empty set
x9 is set
|[(V `1),0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
{|[(V `1),0]|} is functional non empty trivial finite V37() 1 -element Element of bool the carrier of (TOP-REAL 2)
{ ((Ball (|[(V `1),b₁]|,b₁)) \/ {|[(V `1),0]|}) where b<sub>1</sub> is V28() real ext-real Element of REAL : not b<sub>1</sub> <= 0 } is set
{x} is non empty trivial finite 1 -element Element of bool the carrier of ()
r is V28() real ext-real Element of REAL
|[(V `1),r]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[(V `1),r]|,r) is Element of bool the carrier of (TOP-REAL 2)
(Ball (|[(V `1),r]|,r)) \/ {x} is set
f is non empty V28() real ext-real positive non negative Element of REAL
|[(V `1),f]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[(V `1),f]|,f) is Element of bool the carrier of (TOP-REAL 2)
f is V28() real ext-real rational set
u is V28() real ext-real rational set
|[f,u]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[f,u]| - |[(V `1),f]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[f,u]|,|[(V `1),f]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[f,u]| - |[(V `1),f]|).| is V28() real ext-real non negative Element of REAL
0. (TOP-REAL 2) is Relation-like NAT -defined Function-like finite 2 -element V74( TOP-REAL 2) FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
the ZeroF of (TOP-REAL 2) is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
c is non empty V28() real ext-real positive non negative set
Ball (|[(V `1),f]|,c) is Element of bool the carrier of (TOP-REAL 2)
A is V28() real ext-real rational set
b is V28() real ext-real rational set
|[A,b]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[A,b]| - |[(V `1),f]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[A,b]|,|[(V `1),f]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[A,b]| - |[(V `1),f]|).| is V28() real ext-real non negative Element of REAL
{ ((Ball (|[(V `1),(V `2)]|,b₁)) /\ ()) where b₁ is V28() real ext-real Element of REAL : not b₁ <= 0 } is set
r is V28() real ext-real Element of REAL
Ball (V,r) is Element of bool the carrier of (TOP-REAL 2)
(Ball (V,r)) /\ () is Element of bool the carrier of (TOP-REAL 2)
f is non empty V28() real ext-real positive non negative Element of REAL
f is non empty V28() real ext-real positive non negative Element of REAL
min (f,f) is V28() real ext-real set
u is non empty V28() real ext-real positive non negative Element of REAL
Ball (V,u) is Element of bool the carrier of (TOP-REAL 2)
b is V28() real ext-real rational set
c is V28() real ext-real rational set
|[b,c]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[b,c]| - V is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[b,c]|,V) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[b,c]| - V).| is V28() real ext-real non negative Element of REAL
0. (TOP-REAL 2) is Relation-like NAT -defined Function-like finite 2 -element V74( TOP-REAL 2) FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
the ZeroF of (TOP-REAL 2) is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
b is non empty V28() real ext-real positive non negative set
Ball (V,b) is Element of bool the carrier of (TOP-REAL 2)
x is V28() real ext-real rational set
z is V28() real ext-real rational set
|[x,z]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[x,z]| - V is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[x,z]|,V) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[x,z]| - V).| is V28() real ext-real non negative Element of REAL
Ball (V,f) is Element of bool the carrier of (TOP-REAL 2)
X is Element of bool the carrier of ()
B is set
{B} is non empty trivial finite 1 -element set
X \ {B} is Element of bool the carrier of ()
Cl (X \ {B}) is closed Element of bool the carrier of ()
X /\ (product <*RAT,RAT*>) is Element of bool the carrier of ()
uB is Element of bool the carrier of ()
uB \ {B} is Element of bool the carrier of ()
Cl (uB \ {B}) is closed Element of bool the carrier of ()
X is Element of bool the carrier of ()
Cl X is closed Element of bool the carrier of ()
{X} is non empty trivial finite 1 -element Element of bool (bool the carrier of ())
bool (bool the carrier of ()) is non empty set
X \ {X} is Element of bool the carrier of ()
B is set
X is Element of bool the carrier of ()
Cl X is closed Element of bool the carrier of ()
Int X is open Element of bool the carrier of ()
X ` is Element of bool the carrier of ()
the carrier of () \ X is set
Cl (X `) is closed Element of bool the carrier of ()
(Cl (X `)) ` is open Element of bool the carrier of ()
the carrier of () \ (Cl (X `)) is set
([#] ()) ` is open closed Element of bool the carrier of ()
the carrier of () \ ([#] ()) is set
X is Element of bool the carrier of ()
{X} is non empty trivial finite 1 -element Element of bool (bool the carrier of ())
bool (bool the carrier of ()) is non empty set
X \ {X} is Element of bool the carrier of ()
B is set
Cl X is closed Element of bool the carrier of ()
X is Element of bool the carrier of ()
B is set
Der X is Element of bool the carrier of ()
uB is Element of the carrier of ()
{uB} is non empty trivial finite 1 -element Element of bool the carrier of ()
X \ {uB} is Element of bool the carrier of ()
Cl (X \ {uB}) is closed Element of bool the carrier of ()
X is open Element of bool the carrier of ()
Cl X is closed Element of bool the carrier of ()
X is Element of bool the carrier of ()
B is set
Der X is Element of bool the carrier of ()
uB is Element of the carrier of ()
{uB} is non empty trivial finite 1 -element Element of bool the carrier of ()
X \ {uB} is Element of bool the carrier of ()
Cl (X \ {uB}) is closed Element of bool the carrier of ()
X is Element of bool the carrier of ()
Cl X is closed Element of bool the carrier of ()
Int (Cl X) is open Element of bool the carrier of ()
Int X is open Element of bool the carrier of ()
X is Relation-like the carrier of () -defined Function-like total Neighborhood_System of ()
B is Element of bool the carrier of ()
Der B is Element of bool the carrier of ()
the Element of Der B is Element of Der B
x is Element of the carrier of ()
{x} is non empty trivial finite 1 -element Element of bool the carrier of ()
B \ {x} is Element of bool the carrier of ()
Cl (B \ {x}) is closed Element of bool the carrier of ()
V is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
V `1 is V28() real ext-real Element of REAL
V `2 is V28() real ext-real Element of REAL
|[(V `1),(V `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Cl B is closed Element of bool the carrier of ()
X . x is open V235((),x) Element of bool (bool the carrier of ())
bool (bool the carrier of ()) is non empty set
|[(V `1),0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
{|[(V `1),0]|} is functional non empty trivial finite V37() 1 -element Element of bool the carrier of (TOP-REAL 2)
{ ((Ball (|[(V `1),b<sub>1</sub>]|,b<sub>1</sub>)) \/ {|[(V `1),0]|}) where b₁ is V28() real ext-real Element of REAL : not b₁ <= 0 } is set
|[(V `1),1]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[(V `1),1]|,1) is Element of bool the carrier of (TOP-REAL 2)
{V} is functional non empty trivial finite V37() 1 -element Element of bool the carrier of (TOP-REAL 2)
(Ball (|[(V `1),1]|,1)) \/ {V} is Element of bool the carrier of (TOP-REAL 2)
V9 is set
x9 is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
x9 `1 is V28() real ext-real Element of REAL
x9 `2 is V28() real ext-real Element of REAL
|[(x9 `1),(x9 `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
x9 - |[(V `1),1]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(x9,|[(V `1),1]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
(x9 `1) - (V `1) is V28() real ext-real Element of REAL
0 - 1 is V28() real ext-real rational Element of REAL
|[((x9 `1) - (V `1)),(0 - 1)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[((x9 `1) - (V `1)),(0 - 1)]| `2 is V28() real ext-real Element of REAL
|[((x9 `1) - (V `1)),(0 - 1)]| `1 is V28() real ext-real Element of REAL
|.(x9 - |[(V `1),1]|).| is V28() real ext-real non negative Element of REAL
((x9 `1) - (V `1)) ^2 is V28() real ext-real Element of REAL
((x9 `1) - (V `1)) * ((x9 `1) - (V `1)) is V28() real ext-real set
(- 1) ^2 is V28() real ext-real Element of REAL
(- 1) * (- 1) is non empty V28() real ext-real positive non negative rational set
(((x9 `1) - (V `1)) ^2) + ((- 1) ^2) is V28() real ext-real Element of REAL
sqrt ((((x9 `1) - (V `1)) ^2) + ((- 1) ^2)) is V28() real ext-real Element of REAL
1 ^2 is V28() real ext-real Element of REAL
1 * 1 is non empty V28() real ext-real positive non negative rational set
(((x9 `1) - (V `1)) ^2) + (1 ^2) is V28() real ext-real Element of REAL
sqrt ((((x9 `1) - (V `1)) ^2) + (1 ^2)) is V28() real ext-real Element of REAL
abs 1 is V28() real ext-real Element of REAL
bool () is non empty set
B is Element of bool ()
X is closed Element of bool the carrier of ()
uB is Element of bool the carrier of ()
Der uB is Element of bool the carrier of ()
Der X is Element of bool the carrier of ()
Cl uB is closed Element of bool the carrier of ()
uB \/ {} is set
the topology of Sorgenfrey-line is non empty Element of bool (bool the carrier of Sorgenfrey-line)
bool (bool the carrier of Sorgenfrey-line) is non empty set
B is Element of bool (bool REAL)
UniCl B is Element of bool (bool REAL)
X is Element of bool the carrier of Sorgenfrey-line
Cl X is closed Element of bool the carrier of Sorgenfrey-line
uB is set
V is Element of bool the carrier of Sorgenfrey-line
V9 is V28() real ext-real Element of REAL
x9 is V28() real ext-real Element of REAL
[.V9,x9.[ is V169() V170() V171() V268() V272() Element of bool REAL
x is Element of the carrier of Sorgenfrey-line
r is V28() real ext-real rational set
density Sorgenfrey-line is ordinal cardinal set
X is dense Element of bool the carrier of Sorgenfrey-line
len X is ordinal cardinal set
X is dense Element of bool the carrier of ()
len X is ordinal cardinal set
len (product <*RAT,RAT*>) is ordinal cardinal set
density () is ordinal cardinal set
B is Element of the carrier of ()
uB is Element of the carrier of ()
x is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
V is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
x - V is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(x,V) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
0. (TOP-REAL 2) is Relation-like NAT -defined Function-like finite 2 -element V74( TOP-REAL 2) FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
the ZeroF of (TOP-REAL 2) is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|.(x - V).| is V28() real ext-real non negative Element of REAL
V9 is non empty V28() real ext-real positive non negative Element of REAL
V9 / 2 is non empty V28() real ext-real positive non negative Element of REAL
r is open Element of bool the carrier of ()
x9 is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
f is open Element of bool the carrier of ()
f is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
V - x9 is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(V,x9) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(V - x9).| is V28() real ext-real non negative Element of REAL
x9 - V is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(x9,V) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(x9 - V).| is V28() real ext-real non negative Element of REAL
(V9 / 2) + (V9 / 2) is non empty V28() real ext-real positive non negative Element of REAL
x - x9 is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(x,x9) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(x - x9).| is V28() real ext-real non negative Element of REAL
|.(x - x9).| + |.(x9 - V).| is V28() real ext-real non negative Element of REAL
V - f is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(V,f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(V - f).| is V28() real ext-real non negative Element of REAL
f - V is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(f,V) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(f - V).| is V28() real ext-real non negative Element of REAL
(V9 / 2) + (V9 / 2) is non empty V28() real ext-real positive non negative Element of REAL
x - f is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(x,f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(x - f).| is V28() real ext-real non negative Element of REAL
|.(x - f).| + |.(f - V).| is V28() real ext-real non negative Element of REAL
X is dense Element of bool the carrier of ()
exp (2,(exp (2,omega))) is ordinal cardinal set
len X is ordinal cardinal set
len (product <*RAT,RAT*>) is ordinal cardinal set
exp (2,(len X)) is ordinal cardinal set
bool () is non empty Element of bool (bool ())
bool (bool ()) is non empty set
uB is set
() \ uB is Element of bool the carrier of (TOP-REAL 2)
x is Element of bool ()
V is Element of bool ()
{} () is Relation-like non-empty empty-yielding NAT -defined RAT -valued ordinal natural Function-like one-to-one constant functional empty proper V28() real ext-real non positive non negative finite finite-yielding V37() cardinal {} -element Function-yielding V65() open closed boundary nowhere_dense FinSequence-like FinSubsequence-like FinSequence-membered V159() V160() V161() V162() V169() V170() V171() V172() V173() V174() V175() dense-in-itself perfect scattered V269() V270() V271() V272() Element of bool the carrier of ()
({} ()) /\ X is Relation-like finite V169() V170() V171() V172() V173() V174() V269() V270() V271() Element of bool the carrier of ()
([#] ()) /\ X is Element of bool the carrier of ()
{} () is Relation-like non-empty empty-yielding NAT -defined RAT -valued ordinal natural Function-like one-to-one constant functional empty proper V28() real ext-real non positive non negative finite finite-yielding V37() cardinal {} -element Function-yielding V65() open closed boundary nowhere_dense FinSequence-like FinSubsequence-like FinSequence-membered V159() V160() V161() V162() V169() V170() V171() V172() V173() V174() V175() dense-in-itself perfect scattered V269() V270() V271() V272() Element of bool the carrier of ()
x ` is Element of bool ()
() \ x is set
(x `) ` is Element of bool ()
() \ (x `) is set
V ` is Element of bool ()
() \ V is set
x9 is closed Element of bool the carrier of ()
{} () is Relation-like non-empty empty-yielding NAT -defined RAT -valued ordinal natural Function-like one-to-one constant functional empty proper V28() real ext-real non positive non negative finite finite-yielding V37() cardinal {} -element Function-yielding V65() FinSequence-like FinSubsequence-like FinSequence-membered V159() V160() V161() V162() V169() V170() V171() V172() V173() V174() V175() V269() V270() V271() V272() Element of bool ()
V9 is closed Element of bool the carrier of ()
r is Element of bool the carrier of ()
f is Element of bool the carrier of ()
r /\ X is Element of bool the carrier of ()
uB is Relation-like Function-like set
proj1 uB is set
x is set
V is set
uB . x is set
uB . V is set
V9 is Element of bool ()
x9 is Element of bool ()
r is set
V9 \ x9 is Element of bool ()
x9 \ V9 is Element of bool ()
uB . x9 is set
() \ x9 is Element of bool the carrier of (TOP-REAL 2)
f is open Element of bool the carrier of ()
f /\ X is Element of bool the carrier of ()
f is open Element of bool the carrier of ()
uB . V9 is set
() \ V9 is Element of bool the carrier of (TOP-REAL 2)
u is open Element of bool the carrier of ()
u /\ X is Element of bool the carrier of ()
b is open Element of bool the carrier of ()
V9 ` is Element of bool ()
() \ V9 is set
x9 /\ (V9 `) is Element of bool ()
f /\ b is open Element of bool the carrier of ()
x9 ` is Element of bool ()
() \ x9 is set
V9 /\ (x9 `) is Element of bool ()
u /\ f is open Element of bool the carrier of ()
c is set
A is set
c is set
len (proj1 uB) is ordinal cardinal set
proj2 uB is set
len (proj2 uB) is ordinal cardinal set
bool X is non empty Element of bool (bool X)
bool X is non empty set
bool (bool X) is non empty set
x is set
V is set
uB . V is set
() \ V is Element of bool the carrier of (TOP-REAL 2)
V9 is open Element of bool the carrier of ()
V9 /\ X is Element of bool the carrier of ()
x9 is open Element of bool the carrier of ()
len (bool X) is ordinal non empty cardinal set
len (bool ()) is ordinal non empty cardinal set
exp (2,continuum) is ordinal cardinal set
{1} is non empty trivial finite V37() 1 -element V169() V170() V171() V172() V173() V174() V267() V268() V269() V270() V271() Element of bool REAL
1 / 2 is non empty V28() real ext-real positive non negative rational Element of REAL
x is TopSpace-like TopStruct
the carrier of x is set
bool the carrier of x is non empty set
[: the carrier of x, the carrier of I[01]:] is Relation-like V159() V160() V161() set
bool [: the carrier of x, the carrier of I[01]:] is non empty set
bool the carrier of I[01] is non empty set
X is V28() real ext-real Element of the carrier of I[01]
uB is V28() real ext-real Element of the carrier of I[01]
[.X,uB.[ is V169() V170() V171() V268() V272() Element of bool REAL
B is V28() real ext-real Element of the carrier of I[01]
].uB,B.] is V169() V170() V171() V267() V272() Element of bool REAL
V is V169() V170() V171() Element of bool the carrier of I[01]
V9 is V169() V170() V171() Element of bool the carrier of I[01]
f is Element of the carrier of x
f is Element of bool the carrier of x
f ` is Element of bool the carrier of x
the carrier of x \ f is set
u is Relation-like the carrier of x -defined the carrier of I[01] -valued Function-like total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of x, the carrier of I[01]:]
u . f is V28() real ext-real set
u .: f is V169() V170() V171() Element of bool the carrier of I[01]
x9 is open V169() V170() V171() Element of bool the carrier of I[01]
u " x9 is Element of bool the carrier of x
b is Element of bool the carrier of x
r is open V169() V170() V171() Element of bool the carrier of I[01]
u " r is Element of bool the carrier of x
c is Element of bool the carrier of x
[#] I[01] is non empty non proper open closed dense non boundary V169() V170() V171() Element of bool the carrier of I[01]
dom u is Element of bool the carrier of x
A is set
u . A is V28() real ext-real set
b is V28() real ext-real Element of the carrier of I[01]
[.(- 1),1.] is V169() V170() V171() V272() Element of bool REAL
V is non empty TopSpace-like TopStruct
the carrier of V is non empty set
bool the carrier of V is non empty set
V9 is closed Element of bool the carrier of V
V9 ` is open Element of bool the carrier of V
the carrier of V \ V9 is set
[: the carrier of V, the carrier of I[01]:] is Relation-like non empty V159() V160() V161() set
bool [: the carrier of V, the carrier of I[01]:] is non empty set
x9 is Element of the carrier of V
uB is V28() real ext-real Element of the carrier of I[01]
V --> uB is Relation-like the carrier of V -defined the carrier of I[01] -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of V, the carrier of I[01]:]
the carrier of V --> uB is Relation-like the carrier of V -defined the carrier of I[01] -valued Function-like constant non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of V, the carrier of I[01]:]
{uB} is non empty trivial finite 1 -element V169() V170() V171() V267() V268() V269() V270() V271() set
[: the carrier of V,{uB}:] is Relation-like non empty V159() V160() V161() set
(V --> uB) . x9 is V28() real ext-real Element of the carrier of I[01]
(V --> uB) .: V9 is V169() V170() V171() Element of bool the carrier of I[01]
bool the carrier of I[01] is non empty set
{x9} is non empty trivial finite 1 -element Element of bool the carrier of V
f is non empty closed Element of bool the carrier of V
r is non empty closed Element of bool the carrier of V
f \/ r is closed Element of bool the carrier of V
V | f is non empty strict TopSpace-like SubSpace of V
X is V28() real ext-real Element of the carrier of (Closed-Interval-TSpace ((- 1),1))
(V | f) --> X is Relation-like the carrier of (V | f) -defined the carrier of (Closed-Interval-TSpace ((- 1),1)) -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of (V | f), the carrier of (Closed-Interval-TSpace ((- 1),1)):]
the carrier of (V | f) is non empty set
[: the carrier of (V | f), the carrier of (Closed-Interval-TSpace ((- 1),1)):] is Relation-like non empty V159() V160() V161() set
bool [: the carrier of (V | f), the carrier of (Closed-Interval-TSpace ((- 1),1)):] is non empty set
the carrier of (V | f) --> X is Relation-like the carrier of (V | f) -defined the carrier of (Closed-Interval-TSpace ((- 1),1)) -valued Function-like constant non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of (V | f), the carrier of (Closed-Interval-TSpace ((- 1),1)):]
{X} is non empty trivial finite 1 -element V169() V170() V171() V267() V268() V269() V270() V271() set
[: the carrier of (V | f),{X}:] is Relation-like non empty V159() V160() V161() set
V | r is non empty strict TopSpace-like SubSpace of V
B is V28() real ext-real Element of the carrier of (Closed-Interval-TSpace ((- 1),1))
(V | r) --> B is Relation-like the carrier of (V | r) -defined the carrier of (Closed-Interval-TSpace ((- 1),1)) -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of (V | r), the carrier of (Closed-Interval-TSpace ((- 1),1)):]
the carrier of (V | r) is non empty set
[: the carrier of (V | r), the carrier of (Closed-Interval-TSpace ((- 1),1)):] is Relation-like non empty V159() V160() V161() set
bool [: the carrier of (V | r), the carrier of (Closed-Interval-TSpace ((- 1),1)):] is non empty set
the carrier of (V | r) --> B is Relation-like the carrier of (V | r) -defined the carrier of (Closed-Interval-TSpace ((- 1),1)) -valued Function-like constant non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of (V | r), the carrier of (Closed-Interval-TSpace ((- 1),1)):]
{B} is non empty trivial finite 1 -element V169() V170() V171() V267() V268() V269() V270() V271() set
[: the carrier of (V | r),{B}:] is Relation-like non empty V159() V160() V161() set
((V | f) --> X) +* ((V | r) --> B) is Relation-like Function-like V159() V160() V161() set
r --> B is Relation-like r -defined the carrier of (Closed-Interval-TSpace ((- 1),1)) -valued Function-like constant non empty total quasi_total V159() V160() V161() Element of bool [:r, the carrier of (Closed-Interval-TSpace ((- 1),1)):]
[:r, the carrier of (Closed-Interval-TSpace ((- 1),1)):] is Relation-like non empty V159() V160() V161() set
bool [:r, the carrier of (Closed-Interval-TSpace ((- 1),1)):] is non empty set
[:r,{B}:] is Relation-like non empty V159() V160() V161() set
f --> 1 is Relation-like non-empty f -defined REAL -valued RAT -valued INT -valued Function-like constant non empty total quasi_total V159() V160() V161() V162() Element of bool [:f,REAL:]
[:f,REAL:] is Relation-like non empty non trivial non finite V159() V160() V161() set
bool [:f,REAL:] is non empty non trivial non finite set
[:f,{1}:] is Relation-like RAT -valued INT -valued non empty V159() V160() V161() V162() set
(f --> 1) .: V9 is V169() V170() V171() V172() V173() V174() V269() Element of bool REAL
f is closed Element of bool the carrier of V
V | f is strict TopSpace-like SubSpace of V
the carrier of (V | f) is set
[: the carrier of (V | f), the carrier of (Closed-Interval-TSpace ((- 1),1)):] is Relation-like V159() V160() V161() set
bool [: the carrier of (V | f), the carrier of (Closed-Interval-TSpace ((- 1),1)):] is non empty set
[: the carrier of V, the carrier of (Closed-Interval-TSpace ((- 1),1)):] is Relation-like non empty V159() V160() V161() set
bool [: the carrier of V, the carrier of (Closed-Interval-TSpace ((- 1),1)):] is non empty set
b is Relation-like the carrier of (V | f) -defined the carrier of (Closed-Interval-TSpace ((- 1),1)) -valued Function-like total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of (V | f), the carrier of (Closed-Interval-TSpace ((- 1),1)):]
c is Relation-like the carrier of V -defined the carrier of (Closed-Interval-TSpace ((- 1),1)) -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of V, the carrier of (Closed-Interval-TSpace ((- 1),1)):]
c | f is Relation-like the carrier of V -defined f -defined the carrier of V -defined the carrier of (Closed-Interval-TSpace ((- 1),1)) -valued Function-like V159() V160() V161() Element of bool [: the carrier of V, the carrier of (Closed-Interval-TSpace ((- 1),1)):]
A is Relation-like the carrier of I[01] -defined the carrier of (Closed-Interval-TSpace ((- 1),1)) -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of I[01], the carrier of (Closed-Interval-TSpace ((- 1),1)):]
A . 0 is V28() real ext-real set
A . 1 is V28() real ext-real set
A is Relation-like the carrier of I[01] -defined the carrier of (Closed-Interval-TSpace ((- 1),1)) -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of I[01], the carrier of (Closed-Interval-TSpace ((- 1),1)):]
A . 0 is V28() real ext-real set
A . 1 is V28() real ext-real set
[: the carrier of (Closed-Interval-TSpace ((- 1),1)), the carrier of I[01]:] is Relation-like non empty V159() V160() V161() set
bool [: the carrier of (Closed-Interval-TSpace ((- 1),1)), the carrier of I[01]:] is non empty set
A /" is Relation-like the carrier of (Closed-Interval-TSpace ((- 1),1)) -defined the carrier of I[01] -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of (Closed-Interval-TSpace ((- 1),1)), the carrier of I[01]:]
b is Relation-like the carrier of (Closed-Interval-TSpace ((- 1),1)) -defined the carrier of I[01] -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of (Closed-Interval-TSpace ((- 1),1)), the carrier of I[01]:]
A " is Relation-like Function-like set
b . B is V28() real ext-real Element of the carrier of I[01]
uB is V28() real ext-real Element of the carrier of I[01]
dom ((V | r) --> B) is non empty Element of bool the carrier of (V | r)
bool the carrier of (V | r) is non empty set
b . x9 is V28() real ext-real set
((V | r) --> B) . x9 is V28() real ext-real set
b * c is Relation-like the carrier of V -defined the carrier of V -defined the carrier of I[01] -valued the carrier of I[01] -valued Function-like non empty total total quasi_total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of V, the carrier of I[01]:]
x is Relation-like the carrier of V -defined the carrier of I[01] -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of V, the carrier of I[01]:]
x .: V9 is V169() V170() V171() Element of bool the carrier of I[01]
bool the carrier of I[01] is non empty set
c .: V9 is V169() V170() V171() Element of bool the carrier of (Closed-Interval-TSpace ((- 1),1))
bool the carrier of (Closed-Interval-TSpace ((- 1),1)) is non empty set
b .: (c .: V9) is V169() V170() V171() Element of bool the carrier of I[01]
b .: V9 is V169() V170() V171() Element of bool the carrier of (Closed-Interval-TSpace ((- 1),1))
b .: (b .: V9) is V169() V170() V171() Element of bool the carrier of I[01]
Im (b,1) is V169() V170() V171() set
b .: {1} is finite V169() V170() V171() V269() V270() V271() set
x . x9 is V28() real ext-real set
x . x9 is V28() real ext-real Element of the carrier of I[01]
c . x9 is V28() real ext-real Element of the carrier of (Closed-Interval-TSpace ((- 1),1))
b . (c . x9) is V28() real ext-real Element of the carrier of I[01]
b . X is V28() real ext-real Element of the carrier of I[01]
x is V28() real ext-real Element of the carrier of I[01]
X is TopSpace-like T_0 T_1 TopStruct
the carrier of X is set
bool the carrier of X is non empty set
bool (bool the carrier of X) is non empty set
[: the carrier of X, the carrier of I[01]:] is Relation-like V159() V160() V161() set
bool [: the carrier of X, the carrier of I[01]:] is non empty set
B is open quasi_prebasis Element of bool (bool the carrier of X)
uB is Element of the carrier of X
x is Element of bool the carrier of X
x ` is Element of bool the carrier of X
the carrier of X \ x is set
(x `) ` is Element of bool the carrier of X
the carrier of X \ (x `) is set
X is TopSpace-like TopStruct
the carrier of X is set
bool the carrier of X is non empty set
B is non empty TopSpace-like V229() SubSpace of R^1
the carrier of B is non empty V169() V170() V171() set
[: the carrier of X, the carrier of B:] is Relation-like V159() V160() V161() set
bool [: the carrier of X, the carrier of B:] is non empty set
uB is Relation-like the carrier of X -defined the carrier of B -valued Function-like total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of X, the carrier of B:]
x is Relation-like the carrier of X -defined the carrier of B -valued Function-like total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of X, the carrier of B:]
V is Element of bool the carrier of X
the topology of X is non empty Element of bool (bool the carrier of X)
bool (bool the carrier of X) is non empty set
V9 is Element of the carrier of X
uB . V9 is V28() real ext-real set
x . V9 is V28() real ext-real set
(uB . V9) - (x . V9) is V28() real ext-real set
((uB . V9) - (x . V9)) / 2 is V28() real ext-real Element of REAL
(uB . V9) - (((uB . V9) - (x . V9)) / 2) is V28() real ext-real Element of REAL
(uB . V9) + (((uB . V9) - (x . V9)) / 2) is V28() real ext-real Element of REAL
].((uB . V9) - (((uB . V9) - (x . V9)) / 2)),((uB . V9) + (((uB . V9) - (x . V9)) / 2)).[ is V169() V170() V171() V267() V268() V272() Element of bool REAL
(x . V9) - (((uB . V9) - (x . V9)) / 2) is V28() real ext-real Element of REAL
(x . V9) + (((uB . V9) - (x . V9)) / 2) is V28() real ext-real Element of REAL
].((x . V9) - (((uB . V9) - (x . V9)) / 2)),((x . V9) + (((uB . V9) - (x . V9)) / 2)).[ is V169() V170() V171() V267() V268() V272() Element of bool REAL
bool the carrier of B is non empty set
r is open V169() V170() V171() Element of bool the carrier of R^1
[#] B is non empty non proper open closed dense non boundary V169() V170() V171() Element of bool the carrier of B
r /\ ([#] B) is V169() V170() V171() Element of bool the carrier of B
f is open V169() V170() V171() Element of bool the carrier of R^1
f /\ ([#] B) is V169() V170() V171() Element of bool the carrier of B
u is open V169() V170() V171() Element of bool the carrier of B
x " u is Element of bool the carrier of X
V ` is Element of bool the carrier of X
the carrier of X \ V is set
b is non empty V28() real ext-real positive non negative set
b / 2 is non empty V28() real ext-real positive non negative Element of REAL
(uB . V9) + (b / 2) is V28() real ext-real Element of REAL
f is open V169() V170() V171() Element of bool the carrier of B
uB " f is Element of bool the carrier of X
(uB " f) /\ (x " u) is Element of bool the carrier of X
(x . V9) + (b / 2) is V28() real ext-real Element of REAL
(x . V9) - (b / 2) is V28() real ext-real Element of REAL
(uB . V9) - (b / 2) is V28() real ext-real Element of REAL
c is Element of bool the carrier of X
A is set
x . A is V28() real ext-real set
uB . A is V28() real ext-real set
X is TopSpace-like TopStruct
the carrier of X is set
B is non empty TopSpace-like V229() SubSpace of R^1
the carrier of B is non empty V169() V170() V171() set
[: the carrier of X, the carrier of B:] is Relation-like V159() V160() V161() set
bool [: the carrier of X, the carrier of B:] is non empty set
uB is Relation-like the carrier of X -defined the carrier of B -valued Function-like total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of X, the carrier of B:]
x is Relation-like the carrier of X -defined the carrier of B -valued Function-like total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of X, the carrier of B:]
bool the carrier of X is non empty set
V is Element of bool the carrier of X
V9 is Element of bool the carrier of X
the Relation-like the carrier of X -defined the carrier of B -valued Function-like total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of X, the carrier of B:] is Relation-like the carrier of X -defined the carrier of B -valued Function-like total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of X, the carrier of B:]
r is Element of the carrier of X
the Relation-like the carrier of X -defined the carrier of B -valued Function-like total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of X, the carrier of B:] . r is V28() real ext-real set
uB . r is V28() real ext-real set
x . r is V28() real ext-real set
max ((uB . r),(x . r)) is V28() real ext-real set
x9 is Element of the carrier of X
x . x9 is V28() real ext-real set
uB . x9 is V28() real ext-real set
max ((uB . x9),(x . x9)) is V28() real ext-real set
x9 is Element of the carrier of X
uB . x9 is V28() real ext-real set
x . x9 is V28() real ext-real set
max ((uB . x9),(x . x9)) is V28() real ext-real set
x9 is non empty TopSpace-like TopStruct
the carrier of x9 is non empty set
r is Element of the carrier of x9
x . r is V28() real ext-real set
uB . r is V28() real ext-real set
f is Element of the carrier of x9
x . f is V28() real ext-real set
uB . f is V28() real ext-real set
f is Element of the carrier of x9
uB . f is V28() real ext-real set
x . f is V28() real ext-real set
u is Element of the carrier of x9
uB . u is V28() real ext-real set
x . u is V28() real ext-real set
bool the carrier of x9 is non empty set
[: the carrier of x9, the carrier of B:] is Relation-like non empty V159() V160() V161() set
bool [: the carrier of x9, the carrier of B:] is non empty set
r is non empty closed Element of bool the carrier of x9
x9 | r is non empty strict TopSpace-like SubSpace of x9
the carrier of (x9 | r) is non empty set
[#] (x9 | r) is non empty non proper open closed dense non boundary Element of bool the carrier of (x9 | r)
bool the carrier of (x9 | r) is non empty set
f is Relation-like the carrier of x9 -defined the carrier of B -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of x9, the carrier of B:]
dom f is non empty Element of bool the carrier of x9
f | r is Relation-like the carrier of x9 -defined r -defined the carrier of x9 -defined the carrier of B -valued Function-like V159() V160() V161() Element of bool [: the carrier of x9, the carrier of B:]
dom (f | r) is Element of bool r
bool r is non empty set
[: the carrier of (x9 | r), the carrier of B:] is Relation-like non empty V159() V160() V161() set
bool [: the carrier of (x9 | r), the carrier of B:] is non empty set
f is non empty closed Element of bool the carrier of x9
x9 | f is non empty strict TopSpace-like SubSpace of x9
the carrier of (x9 | f) is non empty set
[#] (x9 | f) is non empty non proper open closed dense non boundary Element of bool the carrier of (x9 | f)
bool the carrier of (x9 | f) is non empty set
u is Relation-like the carrier of x9 -defined the carrier of B -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of x9, the carrier of B:]
dom u is non empty Element of bool the carrier of x9
u | f is Relation-like the carrier of x9 -defined f -defined the carrier of x9 -defined the carrier of B -valued Function-like V159() V160() V161() Element of bool [: the carrier of x9, the carrier of B:]
dom (u | f) is Element of bool f
bool f is non empty set
[: the carrier of (x9 | f), the carrier of B:] is Relation-like non empty V159() V160() V161() set
bool [: the carrier of (x9 | f), the carrier of B:] is non empty set
c is Relation-like the carrier of (x9 | f) -defined the carrier of B -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of (x9 | f), the carrier of B:]
dom c is non empty Element of bool the carrier of (x9 | f)
r \/ f is closed Element of bool the carrier of x9
A is set
x . A is V28() real ext-real set
uB . A is V28() real ext-real set
x9 | (r \/ f) is strict TopSpace-like SubSpace of x9
[#] x9 is non empty non proper open closed dense non boundary Element of bool the carrier of x9
x9 | ([#] x9) is non empty strict TopSpace-like SubSpace of x9
the topology of X is non empty Element of bool (bool the carrier of X)
bool (bool the carrier of X) is non empty set
TopStruct(# the carrier of X, the topology of X #) is strict TopSpace-like TopStruct
the topology of B is non empty Element of bool (bool the carrier of B)
bool the carrier of B is non empty set
bool (bool the carrier of B) is non empty set
TopStruct(# the carrier of B, the topology of B #) is non empty strict TopSpace-like TopStruct
b is Relation-like the carrier of (x9 | r) -defined the carrier of B -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of (x9 | r), the carrier of B:]
dom b is non empty Element of bool the carrier of (x9 | r)
A is set
proj1 b is non empty set
proj1 c is non empty set
(proj1 b) /\ (proj1 c) is set
b . A is V28() real ext-real set
c . A is V28() real ext-real set
(dom b) /\ (dom c) is Element of bool the carrier of (x9 | f)
x . A is V28() real ext-real set
uB . A is V28() real ext-real set
b +* c is Relation-like Function-like V159() V160() V161() set
the carrier of (x9 | (r \/ f)) is set
[: the carrier of (x9 | (r \/ f)), the carrier of B:] is Relation-like V159() V160() V161() set
bool [: the carrier of (x9 | (r \/ f)), the carrier of B:] is non empty set
A is Relation-like the carrier of X -defined the carrier of B -valued Function-like total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of X, the carrier of B:]
b is Element of the carrier of X
A . b is V28() real ext-real set
uB . b is V28() real ext-real set
x . b is V28() real ext-real set
max ((uB . b),(x . b)) is V28() real ext-real set
b . b is V28() real ext-real set
c . b is V28() real ext-real set
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
B is non empty TopSpace-like V229() SubSpace of R^1
the carrier of B is non empty V169() V170() V171() set
[: the carrier of X, the carrier of B:] is Relation-like non empty V159() V160() V161() set
bool [: the carrier of X, the carrier of B:] is non empty set
uB is non empty finite set
x is Relation-like uB -defined Function-like total Function-yielding V65() set
commute x is Relation-like Function-like Function-yielding V65() set
uncurry x is Relation-like Function-like set
curry' (uncurry x) is Relation-like Function-like set
x | {} is Relation-like {} -defined uB -defined Function-like Function-yielding V65() set
commute (x | {}) is Relation-like Function-like Function-yielding V65() set
uncurry (x | {}) is Relation-like Function-like set
curry' (uncurry (x | {})) is Relation-like Function-like set
dom x is finite Element of bool uB
bool uB is non empty finite V37() set
proj2 x is set
Funcs ( the carrier of X, the carrier of B) is functional non empty FUNCTION_DOMAIN of the carrier of X, the carrier of B
V is set
V9 is set
x . V9 is Relation-like Function-like set
Funcs (uB,(Funcs ( the carrier of X, the carrier of B))) is functional non empty FUNCTION_DOMAIN of uB, Funcs ( the carrier of X, the carrier of B)
V is set
x . V is Relation-like Function-like set
V9 is set
x | V9 is Relation-like V9 -defined uB -defined Function-like Function-yielding V65() set
commute (x | V9) is Relation-like Function-like Function-yielding V65() set
uncurry (x | V9) is Relation-like Function-like set
curry' (uncurry (x | V9)) is Relation-like Function-like set
{V} is non empty trivial finite 1 -element set
V9 \/ {V} is set
x | (V9 \/ {V}) is Relation-like V9 \/ {V} -defined uB -defined Function-like Function-yielding V65() set
commute (x | (V9 \/ {V})) is Relation-like Function-like Function-yielding V65() set
uncurry (x | (V9 \/ {V})) is Relation-like Function-like set
curry' (uncurry (x | (V9 \/ {V}))) is Relation-like Function-like set
x9 is Relation-like the carrier of X -defined the carrier of B -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of X, the carrier of B:]
r is Element of the carrier of X
(commute (x | (V9 \/ {V}))) . r is Relation-like Function-like set
proj2 ((commute (x | (V9 \/ {V}))) . r) is set
x9 . r is V28() real ext-real Element of the carrier of B
f is non empty finite V169() V170() V171() V267() V268() V269() V270() V271() Element of bool REAL
sup f is V28() real ext-real set
dom x9 is non empty Element of bool the carrier of X
bool the carrier of X is non empty set
x | {V} is Relation-like {V} -defined uB -defined Function-like finite Function-yielding V65() set
V .--> (x . V) is Relation-like {V} -defined Function-like one-to-one finite Function-yielding V65() set
{V} --> (x . V) is Relation-like {V} -defined {(x . V)} -valued Function-like constant non empty total quasi_total finite Function-yielding V65() Element of bool [:{V},{(x . V)}:]
{(x . V)} is functional non empty trivial finite 1 -element set
[:{V},{(x . V)}:] is Relation-like non empty finite set
bool [:{V},{(x . V)}:] is non empty finite V37() set
commute (x | {V}) is Relation-like Function-like Function-yielding V65() set
uncurry (x | {V}) is Relation-like Function-like set
curry' (uncurry (x | {V})) is Relation-like Function-like set
(commute (x | {V})) . r is Relation-like Function-like set
V .--> (x9 . r) is Relation-like {V} -defined the carrier of B -valued Function-like one-to-one finite V159() V160() V161() set
{V} --> (x9 . r) is Relation-like {V} -defined the carrier of B -valued {(x9 . r)} -valued Function-like constant non empty total quasi_total finite V159() V160() V161() Element of bool [:{V},{(x9 . r)}:]
{(x9 . r)} is non empty trivial finite 1 -element V169() V170() V171() V267() V268() V269() V270() V271() set
[:{V},{(x9 . r)}:] is Relation-like non empty finite V159() V160() V161() set
bool [:{V},{(x9 . r)}:] is non empty finite V37() set
proj2 ((commute (x | {V})) . r) is set
{(x9 . r)} is non empty trivial finite 1 -element V169() V170() V171() V267() V268() V269() V270() V271() Element of bool the carrier of B
bool the carrier of B is non empty set
r is non empty set
x | r is Relation-like r -defined uB -defined Function-like Function-yielding V65() set
commute (x | r) is Relation-like Function-like Function-yielding V65() set
uncurry (x | r) is Relation-like Function-like set
curry' (uncurry (x | r)) is Relation-like Function-like set
f is Relation-like the carrier of X -defined the carrier of B -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of X, the carrier of B:]
f is Relation-like the carrier of X -defined the carrier of B -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of X, the carrier of B:]
x9 is Relation-like the carrier of X -defined the carrier of B -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of X, the carrier of B:]
f is Relation-like the carrier of X -defined the carrier of B -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of X, the carrier of B:]
{V} is non empty trivial finite 1 -element set
V9 \/ {V} is set
x | (V9 \/ {V}) is Relation-like V9 \/ {V} -defined uB -defined Function-like Function-yielding V65() set
commute (x | (V9 \/ {V})) is Relation-like Function-like Function-yielding V65() set
uncurry (x | (V9 \/ {V})) is Relation-like Function-like set
curry' (uncurry (x | (V9 \/ {V}))) is Relation-like Function-like set
[:uB,(Funcs ( the carrier of X, the carrier of B)):] is Relation-like non empty set
bool [:uB,(Funcs ( the carrier of X, the carrier of B)):] is non empty set
[:V9,(Funcs ( the carrier of X, the carrier of B)):] is Relation-like set
bool [:V9,(Funcs ( the carrier of X, the carrier of B)):] is non empty set
Funcs (V9,(Funcs ( the carrier of X, the carrier of B))) is functional non empty FUNCTION_DOMAIN of V9, Funcs ( the carrier of X, the carrier of B)
Funcs (V9, the carrier of B) is functional non empty FUNCTION_DOMAIN of V9, the carrier of B
Funcs ( the carrier of X,(Funcs (V9, the carrier of B))) is functional non empty FUNCTION_DOMAIN of the carrier of X, Funcs (V9, the carrier of B)
[: the carrier of X,(Funcs (V9, the carrier of B)):] is Relation-like non empty set
bool [: the carrier of X,(Funcs (V9, the carrier of B)):] is non empty set
b is Element of the carrier of X
(commute (x | (V9 \/ {V}))) . b is Relation-like Function-like set
proj2 ((commute (x | (V9 \/ {V}))) . b) is set
f . b is V28() real ext-real Element of the carrier of B
c is non empty finite V169() V170() V171() V267() V268() V269() V270() V271() Element of bool REAL
sup c is V28() real ext-real set
[:r, the carrier of B:] is Relation-like non empty V159() V160() V161() set
bool [:r, the carrier of B:] is non empty set
u is Relation-like the carrier of X -defined Funcs (V9, the carrier of B) -valued Function-like non empty total quasi_total Function-yielding V65() Element of bool [: the carrier of X,(Funcs (V9, the carrier of B)):]
u . b is Relation-like V9 -defined the carrier of B -valued Function-like total quasi_total Element of Funcs (V9, the carrier of B)
dom x9 is non empty Element of bool the carrier of X
bool the carrier of X is non empty set
x | {V} is Relation-like {V} -defined uB -defined Function-like finite Function-yielding V65() set
(x | V9) \/ (x | {V}) is Relation-like uB -defined set
commute (x | {V}) is Relation-like Function-like Function-yielding V65() set
uncurry (x | {V}) is Relation-like Function-like set
curry' (uncurry (x | {V})) is Relation-like Function-like set
(commute (x | {V})) . b is Relation-like Function-like set
(u . b) \/ ((commute (x | {V})) . b) is Relation-like set
A is Relation-like r -defined the carrier of B -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [:r, the carrier of B:]
rng A is non empty V169() V170() V171() Element of bool the carrier of B
bool the carrier of B is non empty set
proj2 ((commute (x | {V})) . b) is set
(rng A) \/ (proj2 ((commute (x | {V})) . b)) is set
V .--> (x . V) is Relation-like {V} -defined Function-like one-to-one finite Function-yielding V65() set
{V} --> (x . V) is Relation-like {V} -defined {(x . V)} -valued Function-like constant non empty total quasi_total finite Function-yielding V65() Element of bool [:{V},{(x . V)}:]
{(x . V)} is functional non empty trivial finite 1 -element set
[:{V},{(x . V)}:] is Relation-like non empty finite set
bool [:{V},{(x . V)}:] is non empty finite V37() set
x9 . b is V28() real ext-real Element of the carrier of B
V .--> (x9 . b) is Relation-like {V} -defined the carrier of B -valued Function-like one-to-one finite V159() V160() V161() set
{V} --> (x9 . b) is Relation-like {V} -defined the carrier of B -valued {(x9 . b)} -valued Function-like constant non empty total quasi_total finite V159() V160() V161() Element of bool [:{V},{(x9 . b)}:]
{(x9 . b)} is non empty trivial finite 1 -element V169() V170() V171() V267() V268() V269() V270() V271() set
[:{V},{(x9 . b)}:] is Relation-like non empty finite V159() V160() V161() set
bool [:{V},{(x9 . b)}:] is non empty finite V37() set
b is non empty finite V169() V170() V171() V267() V268() V269() V270() V271() Element of bool REAL
{(x9 . b)} is non empty trivial finite 1 -element V169() V170() V171() V267() V268() V269() V270() V271() Element of bool the carrier of B
b \/ {(x9 . b)} is finite V169() V170() V171() V269() V270() V271() set
f . b is V28() real ext-real Element of the carrier of B
sup b is V28() real ext-real set
sup {(x9 . b)} is V28() real ext-real set
max ((f . b),(sup {(x9 . b)})) is V28() real ext-real set
max ((f . b),(x9 . b)) is V28() real ext-real set
x | uB is Relation-like uB -defined uB -defined Function-like finite Function-yielding V65() set
commute (x | uB) is Relation-like Function-like Function-yielding V65() set
uncurry (x | uB) is Relation-like Function-like set
curry' (uncurry (x | uB)) is Relation-like Function-like set
V is Relation-like the carrier of X -defined the carrier of B -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of X, the carrier of B:]
B is non empty TopSpace-like T_0 T_1 TopStruct
the carrier of B is non empty set
bool the carrier of B is non empty set
bool (bool the carrier of B) is non empty set
[: the carrier of B, the carrier of I[01]:] is Relation-like non empty V159() V160() V161() set
bool [: the carrier of B, the carrier of I[01]:] is non empty set
uB is open quasi_prebasis Element of bool (bool the carrier of B)
x is closed Element of bool the carrier of B
x ` is open Element of bool the carrier of B
the carrier of B \ x is set
V is Element of the carrier of B
FinMeetCl uB is Element of bool (bool the carrier of B)
V9 is Element of bool the carrier of B
x9 is Element of bool (bool the carrier of B)
Intersect x9 is Element of bool the carrier of B
(x `) ` is closed Element of bool the carrier of B
the carrier of B \ (x `) is set
{} B is Relation-like non-empty empty-yielding NAT -defined RAT -valued ordinal natural Function-like one-to-one constant functional empty proper V28() real ext-real non positive non negative finite finite-yielding V37() cardinal {} -element Function-yielding V65() open closed boundary nowhere_dense FinSequence-like FinSubsequence-like FinSequence-membered V159() V160() V161() V162() V169() V170() V171() V172() V173() V174() V175() dense-in-itself perfect scattered V269() V270() V271() V272() Element of bool the carrier of B
X is V28() real ext-real Element of the carrier of I[01]
B --> X is Relation-like the carrier of B -defined the carrier of I[01] -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of B, the carrier of I[01]:]
the carrier of B --> X is Relation-like the carrier of B -defined the carrier of I[01] -valued Function-like constant non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of B, the carrier of I[01]:]
{X} is non empty trivial finite 1 -element V169() V170() V171() V267() V268() V269() V270() V271() set
[: the carrier of B,{X}:] is Relation-like non empty V159() V160() V161() set
(B --> X) .: x is V169() V170() V171() Element of bool the carrier of I[01]
bool the carrier of I[01] is non empty set
(B --> X) . V is V28() real ext-real Element of the carrier of I[01]
{0} is non empty trivial finite V37() 1 -element V169() V170() V171() V172() V173() V174() V267() V268() V269() V270() V271() Element of bool REAL
r is non empty finite Element of bool (bool the carrier of B)
the Element of r is Element of r
b is set
c is Element of bool the carrier of B
c ` is Element of bool the carrier of B
the carrier of B \ c is set
A is Relation-like the carrier of B -defined the carrier of I[01] -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of B, the carrier of I[01]:]
A . V is V28() real ext-real Element of the carrier of I[01]
A .: (c `) is V169() V170() V171() Element of bool the carrier of I[01]
bool the carrier of I[01] is non empty set
b is Relation-like Function-like set
proj1 b is set
c is set
b . c is set
A is Element of bool the carrier of B
b is Relation-like the carrier of B -defined the carrier of I[01] -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of B, the carrier of I[01]:]
b . V is V28() real ext-real Element of the carrier of I[01]
A ` is Element of bool the carrier of B
the carrier of B \ A is set
b .: (A `) is V169() V170() V171() Element of bool the carrier of I[01]
bool the carrier of I[01] is non empty set
c is Relation-like r -defined Function-like total Function-yielding V65() set
proj2 c is set
Funcs ( the carrier of B, the carrier of I[01]) is functional non empty FUNCTION_DOMAIN of the carrier of B, the carrier of I[01]
A is set
dom c is finite Element of bool r
bool r is non empty finite V37() set
b is set
c . b is Relation-like Function-like set
x is Element of bool the carrier of B
z is Relation-like the carrier of B -defined the carrier of I[01] -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of B, the carrier of I[01]:]
z . V is V28() real ext-real Element of the carrier of I[01]
x ` is Element of bool the carrier of B
the carrier of B \ x is set
z .: (x `) is V169() V170() V171() Element of bool the carrier of I[01]
bool the carrier of I[01] is non empty set
[:r,(Funcs ( the carrier of B, the carrier of I[01])):] is Relation-like non empty set
bool [:r,(Funcs ( the carrier of B, the carrier of I[01])):] is non empty set
Funcs (r,(Funcs ( the carrier of B, the carrier of I[01]))) is functional non empty FUNCTION_DOMAIN of r, Funcs ( the carrier of B, the carrier of I[01])
commute c is Relation-like Function-like Function-yielding V65() set
uncurry c is Relation-like Function-like set
curry' (uncurry c) is Relation-like Function-like set
Funcs (r, the carrier of I[01]) is functional non empty FUNCTION_DOMAIN of r, the carrier of I[01]
Funcs ( the carrier of B,(Funcs (r, the carrier of I[01]))) is functional non empty FUNCTION_DOMAIN of the carrier of B, Funcs (r, the carrier of I[01])
[: the carrier of B,(Funcs (r, the carrier of I[01])):] is Relation-like non empty set
bool [: the carrier of B,(Funcs (r, the carrier of I[01])):] is non empty set
b is set
c . b is Relation-like Function-like set
x is Element of bool the carrier of B
z is Relation-like the carrier of B -defined the carrier of I[01] -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of B, the carrier of I[01]:]
z . V is V28() real ext-real Element of the carrier of I[01]
x ` is Element of bool the carrier of B
the carrier of B \ x is set
z .: (x `) is V169() V170() V171() Element of bool the carrier of I[01]
bool the carrier of I[01] is non empty set
b is Relation-like the carrier of B -defined the carrier of I[01] -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of B, the carrier of I[01]:]
b . V is V28() real ext-real set
b .: x is V169() V170() V171() Element of bool the carrier of I[01]
bool the carrier of I[01] is non empty set
[:r, the carrier of I[01]:] is Relation-like non empty V159() V160() V161() set
bool [:r, the carrier of I[01]:] is non empty set
A is Relation-like the carrier of B -defined Funcs (r, the carrier of I[01]) -valued Function-like non empty total quasi_total Function-yielding V65() Element of bool [: the carrier of B,(Funcs (r, the carrier of I[01])):]
A . V is Relation-like r -defined the carrier of I[01] -valued Function-like total quasi_total Element of Funcs (r, the carrier of I[01])
x is Relation-like r -defined the carrier of I[01] -valued Function-like non empty total quasi_total finite V159() V160() V161() Element of bool [:r, the carrier of I[01]:]
dom x is non empty finite Element of bool r
bool r is non empty finite V37() set
f is non empty finite V169() V170() V171() V267() V268() V269() V270() V271() Element of bool REAL
(commute c) . V is Relation-like Function-like set
proj2 ((commute c) . V) is set
z is set
c . the Element of r is Relation-like Function-like set
((commute c) . V) . the Element of r is set
x is Element of bool the carrier of B
x is Relation-like the carrier of B -defined the carrier of I[01] -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of B, the carrier of I[01]:]
x . V is V28() real ext-real Element of the carrier of I[01]
x ` is Element of bool the carrier of B
the carrier of B \ x is set
x .: (x `) is V169() V170() V171() Element of bool the carrier of I[01]
z is set
x is set
x . x is V28() real ext-real set
c . x is Relation-like Function-like set
x is Element of bool the carrier of B
w is Relation-like the carrier of B -defined the carrier of I[01] -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of B, the carrier of I[01]:]
w . V is V28() real ext-real Element of the carrier of I[01]
x ` is Element of bool the carrier of B
the carrier of B \ x is set
w .: (x `) is V169() V170() V171() Element of bool the carrier of I[01]
b . V is V28() real ext-real Element of the carrier of I[01]
sup f is V28() real ext-real set
z is set
dom b is non empty Element of bool the carrier of B
x is set
b . x is V28() real ext-real set
x is Element of the carrier of B
w is set
A . x is Relation-like r -defined the carrier of I[01] -valued Function-like total quasi_total Element of Funcs (r, the carrier of I[01])
cGx is Relation-like r -defined the carrier of I[01] -valued Function-like non empty total quasi_total finite V159() V160() V161() Element of bool [:r, the carrier of I[01]:]
rng cGx is non empty finite V169() V170() V171() V267() V268() V269() V270() V271() Element of bool the carrier of I[01]
b . x is V28() real ext-real Element of the carrier of I[01]
S is non empty finite V169() V170() V171() V267() V268() V269() V270() V271() Element of bool REAL
sup S is V28() real ext-real set
c . w is Relation-like Function-like set
T is Element of bool the carrier of B
g is Relation-like the carrier of B -defined the carrier of I[01] -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of B, the carrier of I[01]:]
g . V is V28() real ext-real Element of the carrier of I[01]
T ` is Element of bool the carrier of B
the carrier of B \ T is set
g .: (T `) is V169() V170() V171() Element of bool the carrier of I[01]
g . x is V28() real ext-real Element of the carrier of I[01]
cGx . w is V28() real ext-real set
dom cGx is non empty finite Element of bool r
c₂₅ is ext-real set
the topology of Sorgenfrey-line is non empty Element of bool (bool the carrier of Sorgenfrey-line)
bool (bool the carrier of Sorgenfrey-line) is non empty set
B is Element of bool (bool REAL)
UniCl B is Element of bool (bool REAL)
uB is Element of the carrier of Sorgenfrey-line
x is Element of the carrier of Sorgenfrey-line
V9 is V28() real ext-real Element of REAL
V is V28() real ext-real Element of REAL
V9 + 1 is V28() real ext-real Element of REAL
x9 is V28() real ext-real rational set
[.V9,x9.[ is V169() V170() V171() V268() V272() Element of bool REAL
r is V28() real ext-real rational set
[.V,r.[ is V169() V170() V171() V268() V272() Element of bool REAL
f is open Element of bool the carrier of Sorgenfrey-line
f is open Element of bool the carrier of Sorgenfrey-line
V is V28() real ext-real Element of REAL
V9 is V28() real ext-real Element of REAL
V + 1 is V28() real ext-real Element of REAL
x9 is V28() real ext-real rational set
[.V,x9.[ is V169() V170() V171() V268() V272() Element of bool REAL
r is V28() real ext-real rational set
[.V9,r.[ is V169() V170() V171() V268() V272() Element of bool REAL
f is open Element of bool the carrier of Sorgenfrey-line
f is open Element of bool the carrier of Sorgenfrey-line
V9 is V28() real ext-real Element of REAL
V is V28() real ext-real Element of REAL
X is V28() real ext-real set
left_open_halfline X is V169() V170() V171() Element of bool REAL
K200(-infty,X) is V267() V268() V272() set
right_closed_halfline X is V169() V170() V171() Element of bool REAL
K198(X,+infty) is V268() V272() set
uB is open Element of bool the carrier of Sorgenfrey-line
uB ` is closed Element of bool the carrier of Sorgenfrey-line
the carrier of Sorgenfrey-line \ uB is set
X is V28() real ext-real set
left_closed_halfline X is V169() V170() V171() Element of bool REAL
K199(-infty,X) is V267() V272() set
right_open_halfline X is V169() V170() V171() Element of bool REAL
K200(X,+infty) is V267() V268() V272() set
(left_closed_halfline X) ` is V169() V170() V171() Element of bool REAL
REAL \ (left_closed_halfline X) is V169() V170() V171() set
((left_closed_halfline X) `) ` is V169() V170() V171() Element of bool REAL
REAL \ ((left_closed_halfline X) `) is V169() V170() V171() set
uB is open Element of bool the carrier of Sorgenfrey-line
uB ` is closed Element of bool the carrier of Sorgenfrey-line
the carrier of Sorgenfrey-line \ uB is set
X is V28() real ext-real set
right_closed_halfline X is V169() V170() V171() Element of bool REAL
K198(X,+infty) is V268() V272() set
left_open_halfline X is V169() V170() V171() Element of bool REAL
K200(-infty,X) is V267() V268() V272() set
(right_closed_halfline X) ` is V169() V170() V171() Element of bool REAL
REAL \ (right_closed_halfline X) is V169() V170() V171() set
((right_closed_halfline X) `) ` is V169() V170() V171() Element of bool REAL
REAL \ ((right_closed_halfline X) `) is V169() V170() V171() set
uB is open Element of bool the carrier of Sorgenfrey-line
uB ` is closed Element of bool the carrier of Sorgenfrey-line
the carrier of Sorgenfrey-line \ uB is set
X is V28() real ext-real set
B is V28() real ext-real set
[.X,B.[ is V169() V170() V171() V268() V272() Element of bool REAL
right_closed_halfline X is V169() V170() V171() Element of bool REAL
K198(X,+infty) is V268() V272() set
left_open_halfline B is V169() V170() V171() Element of bool REAL
K200(-infty,B) is V267() V268() V272() set
[.X,B.[ ` is V169() V170() V171() Element of bool REAL
REAL \ [.X,B.[ is V169() V170() V171() set
([.X,B.[ `) ` is V169() V170() V171() Element of bool REAL
REAL \ ([.X,B.[ `) is V169() V170() V171() set
left_open_halfline X is V169() V170() V171() Element of bool REAL
K200(-infty,X) is V267() V268() V272() set
right_closed_halfline B is V169() V170() V171() Element of bool REAL
K198(B,+infty) is V268() V272() set
(left_open_halfline X) \/ (right_closed_halfline B) is V169() V170() V171() Element of bool REAL
((left_open_halfline X) \/ (right_closed_halfline B)) ` is V169() V170() V171() Element of bool REAL
REAL \ ((left_open_halfline X) \/ (right_closed_halfline B)) is V169() V170() V171() set
(left_open_halfline X) ` is V169() V170() V171() Element of bool REAL
REAL \ (left_open_halfline X) is V169() V170() V171() set
(right_closed_halfline B) ` is V169() V170() V171() Element of bool REAL
REAL \ (right_closed_halfline B) is V169() V170() V171() set
((left_open_halfline X) `) /\ ((right_closed_halfline B) `) is V169() V170() V171() Element of bool REAL
x is closed Element of bool the carrier of Sorgenfrey-line
x ` is open Element of bool the carrier of Sorgenfrey-line
the carrier of Sorgenfrey-line \ x is set
(x `) ` is closed Element of bool the carrier of Sorgenfrey-line
the carrier of Sorgenfrey-line \ (x `) is set
((x `) `) /\ ((right_closed_halfline B) `) is V169() V170() V171() Element of bool REAL
V is closed Element of bool the carrier of Sorgenfrey-line
x /\ V is closed Element of bool the carrier of Sorgenfrey-line
[: the carrier of Sorgenfrey-line, the carrier of I[01]:] is Relation-like non empty V159() V160() V161() set
bool [: the carrier of Sorgenfrey-line, the carrier of I[01]:] is non empty set
uB is V28() real ext-real set
V is V28() real ext-real rational set
[.uB,V.[ is V169() V170() V171() V268() V272() Element of bool REAL
V9 is open closed Element of bool the carrier of Sorgenfrey-line
Sorgenfrey-line | V9 is strict TopSpace-like SubSpace of Sorgenfrey-line
the carrier of (Sorgenfrey-line | V9) is set
[: the carrier of (Sorgenfrey-line | V9), the carrier of I[01]:] is Relation-like V159() V160() V161() set
bool [: the carrier of (Sorgenfrey-line | V9), the carrier of I[01]:] is non empty set
X is V28() real ext-real Element of the carrier of I[01]
(Sorgenfrey-line | V9) --> X is Relation-like the carrier of (Sorgenfrey-line | V9) -defined the carrier of I[01] -valued Function-like total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of (Sorgenfrey-line | V9), the carrier of I[01]:]
the carrier of (Sorgenfrey-line | V9) --> X is Relation-like the carrier of (Sorgenfrey-line | V9) -defined the carrier of I[01] -valued Function-like constant total quasi_total V159() V160() V161() Element of bool [: the carrier of (Sorgenfrey-line | V9), the carrier of I[01]:]
{X} is non empty trivial finite 1 -element V169() V170() V171() V267() V268() V269() V270() V271() set
[: the carrier of (Sorgenfrey-line | V9),{X}:] is Relation-like V159() V160() V161() set
V9 ` is open closed Element of bool the carrier of Sorgenfrey-line
the carrier of Sorgenfrey-line \ V9 is set
Sorgenfrey-line | (V9 `) is strict TopSpace-like SubSpace of Sorgenfrey-line
the carrier of (Sorgenfrey-line | (V9 `)) is set
[: the carrier of (Sorgenfrey-line | (V9 `)), the carrier of I[01]:] is Relation-like V159() V160() V161() set
bool [: the carrier of (Sorgenfrey-line | (V9 `)), the carrier of I[01]:] is non empty set
B is V28() real ext-real Element of the carrier of I[01]
(Sorgenfrey-line | (V9 `)) --> B is Relation-like the carrier of (Sorgenfrey-line | (V9 `)) -defined the carrier of I[01] -valued Function-like total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of (Sorgenfrey-line | (V9 `)), the carrier of I[01]:]
the carrier of (Sorgenfrey-line | (V9 `)) --> B is Relation-like the carrier of (Sorgenfrey-line | (V9 `)) -defined the carrier of I[01] -valued Function-like constant total quasi_total V159() V160() V161() Element of bool [: the carrier of (Sorgenfrey-line | (V9 `)), the carrier of I[01]:]
{B} is non empty trivial finite 1 -element V169() V170() V171() V267() V268() V269() V270() V271() set
[: the carrier of (Sorgenfrey-line | (V9 `)),{B}:] is Relation-like V159() V160() V161() set
f is set
f is Relation-like the carrier of Sorgenfrey-line -defined the carrier of I[01] -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of Sorgenfrey-line, the carrier of I[01]:]
x9 is Relation-like the carrier of (Sorgenfrey-line | V9) -defined the carrier of I[01] -valued Function-like total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of (Sorgenfrey-line | V9), the carrier of I[01]:]
dom x9 is Element of bool the carrier of (Sorgenfrey-line | V9)
bool the carrier of (Sorgenfrey-line | V9) is non empty set
r is Relation-like the carrier of (Sorgenfrey-line | (V9 `)) -defined the carrier of I[01] -valued Function-like total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of (Sorgenfrey-line | (V9 `)), the carrier of I[01]:]
dom r is Element of bool the carrier of (Sorgenfrey-line | (V9 `))
bool the carrier of (Sorgenfrey-line | (V9 `)) is non empty set
dom f is non empty Element of bool the carrier of Sorgenfrey-line
[#] Sorgenfrey-line is non empty non proper open closed dense non boundary Element of bool the carrier of Sorgenfrey-line
f is set
(dom x9) \/ (dom r) is set
(V9 `) --> 1 is Relation-like non-empty V9 ` -defined REAL -valued RAT -valued INT -valued Function-like constant total quasi_total V159() V160() V161() V162() Element of bool [:(V9 `),REAL:]
[:(V9 `),REAL:] is Relation-like V159() V160() V161() set
bool [:(V9 `),REAL:] is non empty set
[:(V9 `),{1}:] is Relation-like RAT -valued INT -valued V159() V160() V161() V162() set
((V9 `) --> 1) . f is ordinal natural V28() real ext-real finite cardinal V99() rational set
u is Element of the carrier of Sorgenfrey-line
f . f is V28() real ext-real set
r . f is V28() real ext-real set
x9 . f is V28() real ext-real set
V9 \/ (V9 `) is open closed Element of bool the carrier of Sorgenfrey-line
x9 +* r is Relation-like Function-like V159() V160() V161() set
f is Relation-like the carrier of Sorgenfrey-line -defined the carrier of I[01] -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of Sorgenfrey-line, the carrier of I[01]:]
u is Element of the carrier of Sorgenfrey-line
f . u is V28() real ext-real Element of the carrier of I[01]
the topology of Sorgenfrey-line is non empty Element of bool (bool the carrier of Sorgenfrey-line)
bool (bool the carrier of Sorgenfrey-line) is non empty set
B is Element of bool (bool REAL)
UniCl B is Element of bool (bool REAL)
x is Element of the carrier of Sorgenfrey-line
V is Element of bool the carrier of Sorgenfrey-line
uB is open quasi_prebasis Element of bool (bool the carrier of Sorgenfrey-line)
V9 is V28() real ext-real Element of REAL
x9 is V28() real ext-real Element of REAL
[.V9,x9.[ is V169() V170() V171() V268() V272() Element of bool REAL
r is Relation-like the carrier of Sorgenfrey-line -defined the carrier of I[01] -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of Sorgenfrey-line, the carrier of I[01]:]
f is Relation-like the carrier of Sorgenfrey-line -defined the carrier of I[01] -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of Sorgenfrey-line, the carrier of I[01]:]
f . x is V28() real ext-real Element of the carrier of I[01]
V ` is Element of bool the carrier of Sorgenfrey-line
the carrier of Sorgenfrey-line \ V is set
f .: (V `) is V169() V170() V171() Element of bool the carrier of I[01]
bool the carrier of I[01] is non empty set
f is set
u is Element of the carrier of Sorgenfrey-line
f . u is V28() real ext-real Element of the carrier of I[01]
[: the carrier of (), the carrier of I[01]:] is Relation-like non empty V159() V160() V161() set
bool [: the carrier of (), the carrier of I[01]:] is non empty set
X is V28() real ext-real set
|[X,0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
B is non empty V28() real ext-real positive non negative set
|[X,B]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[X,B]|,B) is Element of bool the carrier of (TOP-REAL 2)
2 * B is non empty V28() real ext-real positive non negative Element of REAL
uB is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[X,0]| - uB is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[X,0]|,uB) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[X,0]| - uB).| is V28() real ext-real non negative Element of REAL
|.(|[X,0]| - uB).| ^2 is V28() real ext-real Element of REAL
|.(|[X,0]| - uB).| * |.(|[X,0]| - uB).| is V28() real ext-real non negative set
uB `2 is V28() real ext-real Element of REAL
(2 * B) * (uB `2) is V28() real ext-real Element of REAL
(|.(|[X,0]| - uB).| ^2) / ((2 * B) * (uB `2)) is V28() real ext-real Element of REAL
uB `1 is V28() real ext-real Element of REAL
|[(uB `1),(uB `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
uB - |[X,B]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(uB,|[X,B]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(uB - |[X,B]|).| is V28() real ext-real non negative Element of REAL
(uB `1) - X is V28() real ext-real Element of REAL
(uB `2) - B is V28() real ext-real Element of REAL
|[((uB `1) - X),((uB `2) - B)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|.|[((uB `1) - X),((uB `2) - B)]|.| is V28() real ext-real non negative Element of REAL
B ^2 is V28() real ext-real set
B * B is non empty V28() real ext-real positive non negative set
|.|[((uB `1) - X),((uB `2) - B)]|.| ^2 is V28() real ext-real Element of REAL
|.|[((uB `1) - X),((uB `2) - B)]|.| * |.|[((uB `1) - X),((uB `2) - B)]|.| is V28() real ext-real non negative set
|[((uB `1) - X),((uB `2) - B)]| `2 is V28() real ext-real Element of REAL
|[((uB `1) - X),((uB `2) - B)]| `1 is V28() real ext-real Element of REAL
((uB `1) - X) ^2 is V28() real ext-real Element of REAL
((uB `1) - X) * ((uB `1) - X) is V28() real ext-real set
((uB `2) - B) ^2 is V28() real ext-real Element of REAL
((uB `2) - B) * ((uB `2) - B) is V28() real ext-real set
(((uB `1) - X) ^2) + (((uB `2) - B) ^2) is V28() real ext-real Element of REAL
0 + (B ^2) is V28() real ext-real Element of REAL
(uB `2) ^2 is V28() real ext-real Element of REAL
(uB `2) * (uB `2) is V28() real ext-real set
2 * (uB `2) is V28() real ext-real Element of REAL
(2 * (uB `2)) * B is V28() real ext-real Element of REAL
((uB `2) ^2) - ((2 * (uB `2)) * B) is V28() real ext-real Element of REAL
(((uB `1) - X) ^2) + (((uB `2) ^2) - ((2 * (uB `2)) * B)) is V28() real ext-real Element of REAL
((((uB `1) - X) ^2) + (((uB `2) ^2) - ((2 * (uB `2)) * B))) + (B ^2) is V28() real ext-real Element of REAL
(((uB `1) - X) ^2) + ((uB `2) ^2) is V28() real ext-real Element of REAL
((((uB `1) - X) ^2) + ((uB `2) ^2)) - ((2 * (uB `2)) * B) is V28() real ext-real Element of REAL
(uB `2) - 0 is V28() real ext-real Element of REAL
((uB `2) - 0) ^2 is V28() real ext-real Element of REAL
((uB `2) - 0) * ((uB `2) - 0) is V28() real ext-real set
(((uB `1) - X) ^2) + (((uB `2) - 0) ^2) is V28() real ext-real Element of REAL
|[((uB `1) - X),(uB `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[((uB `1) - X),(uB `2)]| `2 is V28() real ext-real Element of REAL
|[((uB `1) - X),(uB `2)]| `1 is V28() real ext-real Element of REAL
|[((uB `1) - X),((uB `2) - 0)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|.|[((uB `1) - X),((uB `2) - 0)]|.| is V28() real ext-real non negative Element of REAL
|.|[((uB `1) - X),((uB `2) - 0)]|.| ^2 is V28() real ext-real Element of REAL
|.|[((uB `1) - X),((uB `2) - 0)]|.| * |.|[((uB `1) - X),((uB `2) - 0)]|.| is V28() real ext-real non negative set
uB - |[X,0]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(uB,|[X,0]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(uB - |[X,0]|).| is V28() real ext-real non negative Element of REAL
|.(uB - |[X,0]|).| ^2 is V28() real ext-real Element of REAL
|.(uB - |[X,0]|).| * |.(uB - |[X,0]|).| is V28() real ext-real non negative set
uB is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
uB . |[X,0]| is V28() real ext-real set
x is V28() real ext-real set
V is V28() real ext-real non negative set
|[x,V]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
uB . |[x,V]| is V28() real ext-real set
|[X,0]| - |[x,V]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[X,0]|,|[x,V]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[X,0]| - |[x,V]|).| is V28() real ext-real non negative Element of REAL
|.(|[X,0]| - |[x,V]|).| ^2 is V28() real ext-real Element of REAL
|.(|[X,0]| - |[x,V]|).| * |.(|[X,0]| - |[x,V]|).| is V28() real ext-real non negative set
(2 * B) * V is V28() real ext-real non negative Element of REAL
(|.(|[X,0]| - |[x,V]|).| ^2) / ((2 * B) * V) is V28() real ext-real Element of REAL
|[x,V]| `2 is V28() real ext-real Element of REAL
uB is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
uB . |[X,0]| is V28() real ext-real set
x is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
x . |[X,0]| is V28() real ext-real set
V is Element of the carrier of ()
V9 is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
V9 `1 is V28() real ext-real Element of REAL
V9 `2 is V28() real ext-real Element of REAL
|[(V9 `1),(V9 `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
x9 is V28() real ext-real non negative set
uB . V is V28() real ext-real Element of the carrier of I[01]
x . V is V28() real ext-real Element of the carrier of I[01]
x9 is V28() real ext-real non negative set
|[(V9 `1),x9]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
uB . V is V28() real ext-real Element of the carrier of I[01]
x . V is V28() real ext-real Element of the carrier of I[01]
uB . V is V28() real ext-real Element of the carrier of I[01]
|[X,0]| - V9 is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[X,0]|,V9) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[X,0]| - V9).| is V28() real ext-real non negative Element of REAL
|.(|[X,0]| - V9).| ^2 is V28() real ext-real Element of REAL
|.(|[X,0]| - V9).| * |.(|[X,0]| - V9).| is V28() real ext-real non negative set
x9 is V28() real ext-real non negative set
(2 * B) * x9 is V28() real ext-real non negative Element of REAL
(|.(|[X,0]| - V9).| ^2) / ((2 * B) * x9) is V28() real ext-real Element of REAL
x . V is V28() real ext-real Element of the carrier of I[01]
x9 is V28() real ext-real non negative set
|[(V9 `1),x9]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
X `2 is V28() real ext-real Element of REAL
X `1 is V28() real ext-real Element of REAL
x is V28() real ext-real set
|[x,0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
V is non empty V28() real ext-real positive non negative set
(x,V) is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
(x,V) . X is V28() real ext-real set
|[(X `1),(X `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[x,V]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[x,V]|,V) is Element of bool the carrier of (TOP-REAL 2)
2 * V is non empty V28() real ext-real positive non negative Element of REAL
V9 is non empty V28() real ext-real positive non negative set
(2 * V) * V9 is non empty V28() real ext-real positive non negative Element of REAL
0 / ((2 * V) * V9) is V28() real ext-real Element of REAL
|[(X `1),V9]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[x,0]| - |[(X `1),V9]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[x,0]|,|[(X `1),V9]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[x,0]| - |[(X `1),V9]|).| is V28() real ext-real non negative Element of REAL
|.(|[x,0]| - |[(X `1),V9]|).| ^2 is V28() real ext-real Element of REAL
|.(|[x,0]| - |[(X `1),V9]|).| * |.(|[x,0]| - |[(X `1),V9]|).| is V28() real ext-real non negative set
(|.(|[x,0]| - |[(X `1),V9]|).| ^2) / ((2 * V) * V9) is V28() real ext-real Element of REAL
|[x,0]| - X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[x,0]|,X) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[x,0]| - X).| is V28() real ext-real non negative Element of REAL
X is V28() real ext-real set
B is V28() real ext-real set
|[B,0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
uB is non empty V28() real ext-real positive non negative set
(X,uB) is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
(X,uB) . |[B,0]| is V28() real ext-real set
X - B is V28() real ext-real set
|.(X - B).| is V28() real ext-real Element of REAL
- (X - B) is V28() real ext-real set
|[(X - B),uB]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[(X - B),uB]| `2 is V28() real ext-real Element of REAL
|[(X - B),uB]| `1 is V28() real ext-real Element of REAL
|.|[(X - B),uB]|.| is V28() real ext-real non negative Element of REAL
|.|[(X - B),uB]|.| ^2 is V28() real ext-real Element of REAL
|.|[(X - B),uB]|.| * |.|[(X - B),uB]|.| is V28() real ext-real non negative set
uB ^2 is V28() real ext-real set
uB * uB is non empty V28() real ext-real positive non negative set
(X - B) ^2 is V28() real ext-real set
(X - B) * (X - B) is V28() real ext-real set
(uB ^2) + ((X - B) ^2) is V28() real ext-real set
|.(X - B).| ^2 is V28() real ext-real Element of REAL
|.(X - B).| * |.(X - B).| is V28() real ext-real set
uB - 0 is V28() real ext-real Element of REAL
|[(X - B),(uB - 0)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|.|[(X - B),(uB - 0)]|.| is V28() real ext-real non negative Element of REAL
|[X,uB]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[X,uB]| - |[B,0]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[X,uB]|,|[B,0]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[X,uB]| - |[B,0]|).| is V28() real ext-real non negative Element of REAL
|[B,0]| - |[X,uB]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[B,0]|,|[X,uB]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[B,0]| - |[X,uB]|).| is V28() real ext-real non negative Element of REAL
Ball (|[X,uB]|,uB) is Element of bool the carrier of (TOP-REAL 2)
X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
X `2 is V28() real ext-real Element of REAL
X `1 is V28() real ext-real Element of REAL
x is V28() real ext-real set
V is non empty V28() real ext-real positive non negative set
V9 is non empty V28() real ext-real positive non negative set
V9 * V is non empty V28() real ext-real positive non negative set
|[x,(V9 * V)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
X - |[x,(V9 * V)]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(X,|[x,(V9 * V)]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(X - |[x,(V9 * V)]|).| is V28() real ext-real non negative Element of REAL
(x,V9) is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
(x,V9) . X is V28() real ext-real set
(X `1) - x is V28() real ext-real Element of REAL
(X `2) - (V9 * V) is V28() real ext-real Element of REAL
|[((X `1) - x),((X `2) - (V9 * V))]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[((X `1) - x),((X `2) - (V9 * V))]| `1 is V28() real ext-real Element of REAL
|[((X `1) - x),((X `2) - (V9 * V))]| `2 is V28() real ext-real Element of REAL
|[(X `1),(X `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|.|[((X `1) - x),((X `2) - (V9 * V))]|.| is V28() real ext-real non negative Element of REAL
|.|[((X `1) - x),((X `2) - (V9 * V))]|.| ^2 is V28() real ext-real Element of REAL
|.|[((X `1) - x),((X `2) - (V9 * V))]|.| * |.|[((X `1) - x),((X `2) - (V9 * V))]|.| is V28() real ext-real non negative set
(V9 * V) ^2 is V28() real ext-real set
(V9 * V) * (V9 * V) is non empty V28() real ext-real positive non negative set
((X `1) - x) ^2 is V28() real ext-real Element of REAL
((X `1) - x) * ((X `1) - x) is V28() real ext-real set
((X `2) - (V9 * V)) ^2 is V28() real ext-real Element of REAL
((X `2) - (V9 * V)) * ((X `2) - (V9 * V)) is V28() real ext-real set
(((X `1) - x) ^2) + (((X `2) - (V9 * V)) ^2) is V28() real ext-real Element of REAL
(X `2) ^2 is V28() real ext-real Element of REAL
(X `2) * (X `2) is V28() real ext-real set
(((X `1) - x) ^2) + ((X `2) ^2) is V28() real ext-real Element of REAL
2 * (X `2) is V28() real ext-real Element of REAL
(2 * (X `2)) * V9 is V28() real ext-real Element of REAL
((2 * (X `2)) * V9) * V is V28() real ext-real Element of REAL
x9 is non empty V28() real ext-real positive non negative set
x9 - 0 is V28() real ext-real Element of REAL
|[((X `1) - x),(x9 - 0)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[((X `1) - x),(x9 - 0)]| `1 is V28() real ext-real Element of REAL
|[((X `1) - x),x9]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[((X `1) - x),x9]| `2 is V28() real ext-real Element of REAL
V9 - (V9 * V) is V28() real ext-real set
|[x,V9]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
X - |[x,V9]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(X,|[x,V9]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(X - |[x,V9]|).| is V28() real ext-real non negative Element of REAL
|.(X - |[x,V9]|).| ^2 is V28() real ext-real Element of REAL
|.(X - |[x,V9]|).| * |.(X - |[x,V9]|).| is V28() real ext-real non negative set
x9 - V9 is V28() real ext-real set
|[((X `1) - x),(x9 - V9)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|.|[((X `1) - x),(x9 - V9)]|.| is V28() real ext-real non negative Element of REAL
|.|[((X `1) - x),(x9 - V9)]|.| ^2 is V28() real ext-real Element of REAL
|.|[((X `1) - x),(x9 - V9)]|.| * |.|[((X `1) - x),(x9 - V9)]|.| is V28() real ext-real non negative set
(x9 - V9) ^2 is V28() real ext-real set
(x9 - V9) * (x9 - V9) is V28() real ext-real set
(((X `1) - x) ^2) + ((x9 - V9) ^2) is V28() real ext-real Element of REAL
V * V9 is non empty V28() real ext-real positive non negative set
x9 - (V * V9) is V28() real ext-real set
(x9 - (V * V9)) ^2 is V28() real ext-real set
(x9 - (V * V9)) * (x9 - (V * V9)) is V28() real ext-real set
(((X `1) - x) ^2) + ((x9 - (V * V9)) ^2) is V28() real ext-real Element of REAL
V9 - (V * V9) is V28() real ext-real set
(V9 - (V * V9)) ^2 is V28() real ext-real set
(V9 - (V * V9)) * (V9 - (V * V9)) is V28() real ext-real set
2 * (V9 - (V * V9)) is V28() real ext-real Element of REAL
(V * V9) - x9 is V28() real ext-real set
(2 * (V9 - (V * V9))) * ((V * V9) - x9) is V28() real ext-real Element of REAL
((V9 - (V * V9)) ^2) + ((2 * (V9 - (V * V9))) * ((V * V9) - x9)) is V28() real ext-real Element of REAL
((((X `1) - x) ^2) + ((x9 - (V * V9)) ^2)) + (((V9 - (V * V9)) ^2) + ((2 * (V9 - (V * V9))) * ((V * V9) - x9))) is V28() real ext-real Element of REAL
|[((X `1) - x),(x9 - (V * V9))]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|.|[((X `1) - x),(x9 - (V * V9))]|.| is V28() real ext-real non negative Element of REAL
|.|[((X `1) - x),(x9 - (V * V9))]|.| ^2 is V28() real ext-real Element of REAL
|.|[((X `1) - x),(x9 - (V * V9))]|.| * |.|[((X `1) - x),(x9 - (V * V9))]|.| is V28() real ext-real non negative set
(|.|[((X `1) - x),(x9 - (V * V9))]|.| ^2) + (((V9 - (V * V9)) ^2) + ((2 * (V9 - (V * V9))) * ((V * V9) - x9))) is V28() real ext-real Element of REAL
(V * V9) ^2 is V28() real ext-real set
(V * V9) * (V * V9) is non empty V28() real ext-real positive non negative set
V9 * V9 is non empty V28() real ext-real positive non negative set
V9 * x9 is non empty V28() real ext-real positive non negative set
(V9 * V9) - (V9 * x9) is V28() real ext-real set
(V9 * V) * V9 is non empty V28() real ext-real positive non negative set
((V9 * V9) - (V9 * x9)) + ((V9 * V) * V9) is V28() real ext-real set
(((V9 * V9) - (V9 * x9)) + ((V9 * V) * V9)) - (V9 * x9) is V28() real ext-real set
(V * V9) * V9 is non empty V28() real ext-real positive non negative set
(V * V9) * x9 is non empty V28() real ext-real positive non negative set
((V * V9) * V9) - ((V * V9) * x9) is V28() real ext-real set
(((V * V9) * V9) - ((V * V9) * x9)) + ((V * V9) ^2) is V28() real ext-real set
((((V * V9) * V9) - ((V * V9) * x9)) + ((V * V9) ^2)) - ((V * V9) * x9) is V28() real ext-real set
((((V9 * V9) - (V9 * x9)) + ((V9 * V) * V9)) - (V9 * x9)) - (((((V * V9) * V9) - ((V * V9) * x9)) + ((V * V9) ^2)) - ((V * V9) * x9)) is V28() real ext-real set
((V * V9) ^2) + (((((V9 * V9) - (V9 * x9)) + ((V9 * V) * V9)) - (V9 * x9)) - (((((V * V9) * V9) - ((V * V9) * x9)) + ((V * V9) ^2)) - ((V * V9) * x9))) is V28() real ext-real set
V9 ^2 is V28() real ext-real set
1 + 1 is non empty V28() real ext-real positive non negative rational Element of REAL
(1 + 1) * x9 is non empty V28() real ext-real positive non negative Element of REAL
r is non empty V28() real ext-real positive non negative set
((1 + 1) * x9) * r is non empty V28() real ext-real positive non negative Element of REAL
(V9 ^2) - (((1 + 1) * x9) * r) is V28() real ext-real Element of REAL
Ball (|[x,V9]|,V9) is Element of bool the carrier of (TOP-REAL 2)
|[x,0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[x,0]| - X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[x,0]|,X) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[x,0]| - X).| is V28() real ext-real non negative Element of REAL
|.(|[x,0]| - X).| ^2 is V28() real ext-real Element of REAL
|.(|[x,0]| - X).| * |.(|[x,0]| - X).| is V28() real ext-real non negative set
2 * V9 is non empty V28() real ext-real positive non negative Element of REAL
(2 * V9) * x9 is non empty V28() real ext-real positive non negative Element of REAL
(|.(|[x,0]| - X).| ^2) / ((2 * V9) * x9) is V28() real ext-real Element of REAL
X - |[x,0]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(X,|[x,0]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(X - |[x,0]|).| is V28() real ext-real non negative Element of REAL
|.(X - |[x,0]|).| ^2 is V28() real ext-real Element of REAL
|.(X - |[x,0]|).| * |.(X - |[x,0]|).| is V28() real ext-real non negative set
(|.(X - |[x,0]|).| ^2) / ((2 * V9) * x9) is V28() real ext-real Element of REAL
|.|[((X `1) - x),(x9 - 0)]|.| is V28() real ext-real non negative Element of REAL
|.|[((X `1) - x),(x9 - 0)]|.| ^2 is V28() real ext-real Element of REAL
|.|[((X `1) - x),(x9 - 0)]|.| * |.|[((X `1) - x),(x9 - 0)]|.| is V28() real ext-real non negative set
(|.|[((X `1) - x),(x9 - 0)]|.| ^2) / ((2 * V9) * x9) is V28() real ext-real Element of REAL
x9 ^2 is V28() real ext-real set
x9 * x9 is non empty V28() real ext-real positive non negative set
(((X `1) - x) ^2) + (x9 ^2) is V28() real ext-real Element of REAL
((((X `1) - x) ^2) + (x9 ^2)) / ((2 * V9) * x9) is V28() real ext-real Element of REAL
2 * x9 is non empty V28() real ext-real positive non negative Element of REAL
(2 * x9) * V9 is non empty V28() real ext-real positive non negative Element of REAL
((2 * x9) * V9) * V is non empty V28() real ext-real positive non negative Element of REAL
(((2 * x9) * V9) * V) / ((2 * V9) * x9) is non empty V28() real ext-real positive non negative Element of REAL
((2 * x9) * V9) / ((2 * V9) * x9) is non empty V28() real ext-real positive non negative Element of REAL
V * (((2 * x9) * V9) / ((2 * V9) * x9)) is non empty V28() real ext-real positive non negative Element of REAL
V * 1 is non empty V28() real ext-real positive non negative Element of REAL
|[x,V9]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[x,V9]|,V9) is Element of bool the carrier of (TOP-REAL 2)
X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
X `1 is V28() real ext-real Element of REAL
X `2 is V28() real ext-real Element of REAL
V is V28() real ext-real set
x is V28() real ext-real set
V9 is non empty V28() real ext-real positive non negative set
V9 * V is V28() real ext-real set
|[x,(V9 * V)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
X - |[x,(V9 * V)]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(X,|[x,(V9 * V)]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(X - |[x,(V9 * V)]|).| is V28() real ext-real non negative Element of REAL
(x,V9) is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
(x,V9) . X is V28() real ext-real set
(X `1) - x is V28() real ext-real Element of REAL
(X `2) - (V9 * V) is V28() real ext-real Element of REAL
|[((X `1) - x),((X `2) - (V9 * V))]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[((X `1) - x),((X `2) - (V9 * V))]| `2 is V28() real ext-real Element of REAL
|[(X `1),(X `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
V9 * 1 is non empty V28() real ext-real positive non negative Element of REAL
|[((X `1) - x),((X `2) - (V9 * V))]| `1 is V28() real ext-real Element of REAL
f is non empty V28() real ext-real positive non negative set
|[x,f]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[x,f]|,f) is Element of bool the carrier of (TOP-REAL 2)
(V9 * V) ^2 is V28() real ext-real set
(V9 * V) * (V9 * V) is V28() real ext-real set
|.(X - |[x,(V9 * V)]|).| ^2 is V28() real ext-real Element of REAL
|.(X - |[x,(V9 * V)]|).| * |.(X - |[x,(V9 * V)]|).| is V28() real ext-real non negative set
|.|[((X `1) - x),((X `2) - (V9 * V))]|.| is V28() real ext-real non negative Element of REAL
|.|[((X `1) - x),((X `2) - (V9 * V))]|.| ^2 is V28() real ext-real Element of REAL
|.|[((X `1) - x),((X `2) - (V9 * V))]|.| * |.|[((X `1) - x),((X `2) - (V9 * V))]|.| is V28() real ext-real non negative set
((X `1) - x) ^2 is V28() real ext-real Element of REAL
((X `1) - x) * ((X `1) - x) is V28() real ext-real set
((X `2) - (V9 * V)) ^2 is V28() real ext-real Element of REAL
((X `2) - (V9 * V)) * ((X `2) - (V9 * V)) is V28() real ext-real set
(((X `1) - x) ^2) + (((X `2) - (V9 * V)) ^2) is V28() real ext-real Element of REAL
(X `2) ^2 is V28() real ext-real Element of REAL
(X `2) * (X `2) is V28() real ext-real set
(((X `1) - x) ^2) + ((X `2) ^2) is V28() real ext-real Element of REAL
2 * (X `2) is V28() real ext-real Element of REAL
(2 * (X `2)) * (V9 * V) is V28() real ext-real Element of REAL
((((X `1) - x) ^2) + ((X `2) ^2)) - ((2 * (X `2)) * (V9 * V)) is V28() real ext-real Element of REAL
(((((X `1) - x) ^2) + ((X `2) ^2)) - ((2 * (X `2)) * (V9 * V))) + ((V9 * V) ^2) is V28() real ext-real Element of REAL
(2 * (X `2)) * V9 is V28() real ext-real Element of REAL
((2 * (X `2)) * V9) * V is V28() real ext-real Element of REAL
((((X `1) - x) ^2) + ((X `2) ^2)) - (((2 * (X `2)) * V9) * V) is V28() real ext-real Element of REAL
f is non empty V28() real ext-real positive non negative set
f - 0 is V28() real ext-real Element of REAL
|[((X `1) - x),(f - 0)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[((X `1) - x),(f - 0)]| `1 is V28() real ext-real Element of REAL
|[((X `1) - x),f]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[((X `1) - x),f]| `2 is V28() real ext-real Element of REAL
|[x,(V9 * 1)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[x,(V9 * 1)]|,(V9 * 1)) is Element of bool the carrier of (TOP-REAL 2)
|[x,0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[x,0]| - X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[x,0]|,X) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[x,0]| - X).| is V28() real ext-real non negative Element of REAL
|.(|[x,0]| - X).| ^2 is V28() real ext-real Element of REAL
|.(|[x,0]| - X).| * |.(|[x,0]| - X).| is V28() real ext-real non negative set
2 * V9 is non empty V28() real ext-real positive non negative Element of REAL
(2 * V9) * f is non empty V28() real ext-real positive non negative Element of REAL
(|.(|[x,0]| - X).| ^2) / ((2 * V9) * f) is V28() real ext-real Element of REAL
X - |[x,0]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(X,|[x,0]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(X - |[x,0]|).| is V28() real ext-real non negative Element of REAL
|.(X - |[x,0]|).| ^2 is V28() real ext-real Element of REAL
|.(X - |[x,0]|).| * |.(X - |[x,0]|).| is V28() real ext-real non negative set
(|.(X - |[x,0]|).| ^2) / ((2 * V9) * f) is V28() real ext-real Element of REAL
|.|[((X `1) - x),(f - 0)]|.| is V28() real ext-real non negative Element of REAL
|.|[((X `1) - x),(f - 0)]|.| ^2 is V28() real ext-real Element of REAL
|.|[((X `1) - x),(f - 0)]|.| * |.|[((X `1) - x),(f - 0)]|.| is V28() real ext-real non negative set
(|.|[((X `1) - x),(f - 0)]|.| ^2) / ((2 * V9) * f) is V28() real ext-real Element of REAL
f ^2 is V28() real ext-real set
f * f is non empty V28() real ext-real positive non negative set
(((X `1) - x) ^2) + (f ^2) is V28() real ext-real Element of REAL
((((X `1) - x) ^2) + (f ^2)) / ((2 * V9) * f) is V28() real ext-real Element of REAL
2 * f is non empty V28() real ext-real positive non negative Element of REAL
(2 * f) * V9 is non empty V28() real ext-real positive non negative Element of REAL
((2 * f) * V9) * V is V28() real ext-real Element of REAL
(((2 * f) * V9) * V) / ((2 * V9) * f) is V28() real ext-real Element of REAL
((2 * f) * V9) / ((2 * V9) * f) is non empty V28() real ext-real positive non negative Element of REAL
V * (((2 * f) * V9) / ((2 * V9) * f)) is V28() real ext-real Element of REAL
V * 1 is V28() real ext-real Element of REAL
X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
X `2 is V28() real ext-real Element of REAL
X `1 is V28() real ext-real Element of REAL
V9 is V28() real ext-real set
V is V28() real ext-real set
x9 is non empty V28() real ext-real positive non negative set
x9 * V9 is V28() real ext-real set
|[V,(x9 * V9)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
X - |[V,(x9 * V9)]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(X,|[V,(x9 * V9)]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(X - |[V,(x9 * V9)]|).| is V28() real ext-real non negative Element of REAL
(V,x9) is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
(V,x9) . X is V28() real ext-real set
|[V,0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[V,0]| - |[V,(x9 * V9)]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[V,0]|,|[V,(x9 * V9)]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[V,0]| - |[V,(x9 * V9)]|).| is V28() real ext-real non negative Element of REAL
|[V,(x9 * V9)]| - |[V,0]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[V,(x9 * V9)]|,|[V,0]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[V,(x9 * V9)]| - |[V,0]|).| is V28() real ext-real non negative Element of REAL
V - V is V28() real ext-real set
f is V28() real ext-real Element of REAL
f - 0 is V28() real ext-real Element of REAL
|[(V - V),(f - 0)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|.|[(V - V),(f - 0)]|.| is V28() real ext-real non negative Element of REAL
abs f is V28() real ext-real Element of REAL
r is V28() real ext-real non negative set
x9 * r is V28() real ext-real non negative set
x is V28() real ext-real non negative set
|[(X `1),(X `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
f is Element of the carrier of ()
(V,x9) . f is V28() real ext-real Element of the carrier of I[01]
x9 * 0 is V28() real ext-real Element of REAL
|[V,(x9 * 0)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
(X `1) - V is V28() real ext-real Element of REAL
x - 0 is V28() real ext-real Element of REAL
|[((X `1) - V),(x - 0)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[((X `1) - V),(x - 0)]| `1 is V28() real ext-real Element of REAL
|[((X `1) - V),x]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[((X `1) - V),x]| `2 is V28() real ext-real Element of REAL
|[V,x9]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[V,x9]|,x9) is Element of bool the carrier of (TOP-REAL 2)
|[V,0]| - X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[V,0]|,X) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[V,0]| - X).| is V28() real ext-real non negative Element of REAL
|.(|[V,0]| - X).| ^2 is V28() real ext-real Element of REAL
|.(|[V,0]| - X).| * |.(|[V,0]| - X).| is V28() real ext-real non negative set
2 * x9 is non empty V28() real ext-real positive non negative Element of REAL
(2 * x9) * x is V28() real ext-real non negative Element of REAL
(|.(|[V,0]| - X).| ^2) / ((2 * x9) * x) is V28() real ext-real Element of REAL
X - |[V,0]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(X,|[V,0]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(X - |[V,0]|).| is V28() real ext-real non negative Element of REAL
|.(X - |[V,0]|).| ^2 is V28() real ext-real Element of REAL
|.(X - |[V,0]|).| * |.(X - |[V,0]|).| is V28() real ext-real non negative set
(|.(X - |[V,0]|).| ^2) / ((2 * x9) * x) is V28() real ext-real Element of REAL
|.|[((X `1) - V),(x - 0)]|.| is V28() real ext-real non negative Element of REAL
|.|[((X `1) - V),(x - 0)]|.| ^2 is V28() real ext-real Element of REAL
|.|[((X `1) - V),(x - 0)]|.| * |.|[((X `1) - V),(x - 0)]|.| is V28() real ext-real non negative set
(|.|[((X `1) - V),(x - 0)]|.| ^2) / ((2 * x9) * x) is V28() real ext-real Element of REAL
((X `1) - V) ^2 is V28() real ext-real Element of REAL
((X `1) - V) * ((X `1) - V) is V28() real ext-real set
x ^2 is V28() real ext-real set
x * x is V28() real ext-real non negative set
(((X `1) - V) ^2) + (x ^2) is V28() real ext-real Element of REAL
((((X `1) - V) ^2) + (x ^2)) / ((2 * x9) * x) is V28() real ext-real Element of REAL
|.(X - |[V,(x9 * V9)]|).| ^2 is V28() real ext-real Element of REAL
|.(X - |[V,(x9 * V9)]|).| * |.(X - |[V,(x9 * V9)]|).| is V28() real ext-real non negative set
(x9 * V9) ^2 is V28() real ext-real set
(x9 * V9) * (x9 * V9) is V28() real ext-real set
x - (x9 * V9) is V28() real ext-real set
|[((X `1) - V),(x - (x9 * V9))]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|.|[((X `1) - V),(x - (x9 * V9))]|.| is V28() real ext-real non negative Element of REAL
|.|[((X `1) - V),(x - (x9 * V9))]|.| ^2 is V28() real ext-real Element of REAL
|.|[((X `1) - V),(x - (x9 * V9))]|.| * |.|[((X `1) - V),(x - (x9 * V9))]|.| is V28() real ext-real non negative set
|[((X `1) - V),(x - (x9 * V9))]| `2 is V28() real ext-real Element of REAL
|[((X `1) - V),(x - (x9 * V9))]| `1 is V28() real ext-real Element of REAL
(x - (x9 * V9)) ^2 is V28() real ext-real set
(x - (x9 * V9)) * (x - (x9 * V9)) is V28() real ext-real set
(((X `1) - V) ^2) + ((x - (x9 * V9)) ^2) is V28() real ext-real Element of REAL
2 * x is V28() real ext-real non negative Element of REAL
(2 * x) * (x9 * V9) is V28() real ext-real Element of REAL
((((X `1) - V) ^2) + (x ^2)) - ((2 * x) * (x9 * V9)) is V28() real ext-real Element of REAL
(((((X `1) - V) ^2) + (x ^2)) - ((2 * x) * (x9 * V9))) + ((x9 * V9) ^2) is V28() real ext-real Element of REAL
(2 * x) * x9 is V28() real ext-real non negative Element of REAL
((2 * x) * x9) * V9 is V28() real ext-real Element of REAL
((((X `1) - V) ^2) + (x ^2)) - (((2 * x) * x9) * V9) is V28() real ext-real Element of REAL
u is non empty V28() real ext-real positive non negative set
2 * u is non empty V28() real ext-real positive non negative Element of REAL
(2 * u) * x9 is non empty V28() real ext-real positive non negative Element of REAL
(2 * x9) * u is non empty V28() real ext-real positive non negative Element of REAL
((2 * u) * x9) / ((2 * x9) * u) is non empty V28() real ext-real positive non negative Element of REAL
V9 * (((2 * u) * x9) / ((2 * x9) * u)) is V28() real ext-real Element of REAL
V9 * 1 is V28() real ext-real Element of REAL
((2 * u) * x9) * V9 is V28() real ext-real Element of REAL
(((2 * u) * x9) * V9) / ((2 * x9) * u) is V28() real ext-real Element of REAL
X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
X `2 is V28() real ext-real Element of REAL
uB is V28() real ext-real set
x is V28() real ext-real set
B is V28() real ext-real set
].uB,x.[ is V169() V170() V171() V267() V268() V272() Element of bool REAL
V is non empty V28() real ext-real positive non negative set
(B,V) is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
(B,V) . X is V28() real ext-real set
(B,V) " ].uB,x.[ is Element of bool the carrier of ()
X `1 is V28() real ext-real Element of REAL
|[(X `1),(X `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[B,V]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[B,V]|,V) is Element of bool the carrier of (TOP-REAL 2)
(X `1) - B is V28() real ext-real Element of REAL
|[((X `1) - B),(X `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[((X `1) - B),(X `2)]| `1 is V28() real ext-real Element of REAL
|[B,0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[B,0]| - X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[B,0]|,X) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[B,0]| - X).| is V28() real ext-real non negative Element of REAL
|.(|[B,0]| - X).| ^2 is V28() real ext-real Element of REAL
|.(|[B,0]| - X).| * |.(|[B,0]| - X).| is V28() real ext-real non negative set
2 * V is non empty V28() real ext-real positive non negative Element of REAL
(2 * V) * (X `2) is V28() real ext-real Element of REAL
(|.(|[B,0]| - X).| ^2) / ((2 * V) * (X `2)) is V28() real ext-real Element of REAL
V9 is non empty V28() real ext-real positive non negative set
|[((X `1) - B),V9]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[((X `1) - B),V9]| `2 is V28() real ext-real Element of REAL
(2 * V) * V9 is non empty V28() real ext-real positive non negative Element of REAL
((2 * V) * (X `2)) * uB is V28() real ext-real Element of REAL
X - |[B,0]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(X,|[B,0]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(X - |[B,0]|).| is V28() real ext-real non negative Element of REAL
|.(X - |[B,0]|).| ^2 is V28() real ext-real Element of REAL
|.(X - |[B,0]|).| * |.(X - |[B,0]|).| is V28() real ext-real non negative set
(X `2) - 0 is V28() real ext-real Element of REAL
|[((X `1) - B),((X `2) - 0)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|.|[((X `1) - B),((X `2) - 0)]|.| is V28() real ext-real non negative Element of REAL
|.|[((X `1) - B),((X `2) - 0)]|.| ^2 is V28() real ext-real Element of REAL
|.|[((X `1) - B),((X `2) - 0)]|.| * |.|[((X `1) - B),((X `2) - 0)]|.| is V28() real ext-real non negative set
((X `1) - B) ^2 is V28() real ext-real Element of REAL
((X `1) - B) * ((X `1) - B) is V28() real ext-real set
V9 ^2 is V28() real ext-real set
V9 * V9 is non empty V28() real ext-real positive non negative set
(((X `1) - B) ^2) + (V9 ^2) is V28() real ext-real Element of REAL
((2 * V) * V9) * uB is V28() real ext-real Element of REAL
((((X `1) - B) ^2) + (V9 ^2)) - (((2 * V) * V9) * uB) is V28() real ext-real Element of REAL
V * uB is V28() real ext-real set
(V * uB) ^2 is V28() real ext-real set
(V * uB) * (V * uB) is V28() real ext-real set
(((((X `1) - B) ^2) + (V9 ^2)) - (((2 * V) * V9) * uB)) + ((V * uB) ^2) is V28() real ext-real Element of REAL
V9 - (V * uB) is V28() real ext-real set
(V9 - (V * uB)) ^2 is V28() real ext-real set
(V9 - (V * uB)) * (V9 - (V * uB)) is V28() real ext-real set
(((X `1) - B) ^2) + ((V9 - (V * uB)) ^2) is V28() real ext-real Element of REAL
((2 * V) * (X `2)) * x is V28() real ext-real Element of REAL
((2 * V) * V9) * x is V28() real ext-real Element of REAL
((((X `1) - B) ^2) + (V9 ^2)) - (((2 * V) * V9) * x) is V28() real ext-real Element of REAL
V * x is V28() real ext-real set
(V * x) ^2 is V28() real ext-real set
(V * x) * (V * x) is V28() real ext-real set
(((((X `1) - B) ^2) + (V9 ^2)) - (((2 * V) * V9) * x)) + ((V * x) ^2) is V28() real ext-real Element of REAL
V9 - (V * x) is V28() real ext-real set
(V9 - (V * x)) ^2 is V28() real ext-real set
(V9 - (V * x)) * (V9 - (V * x)) is V28() real ext-real set
(((X `1) - B) ^2) + ((V9 - (V * x)) ^2) is V28() real ext-real Element of REAL
|[((X `1) - B),(V9 - (V * x))]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[((X `1) - B),(V9 - (V * x))]| `2 is V28() real ext-real Element of REAL
|[((X `1) - B),(V9 - (V * uB))]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[((X `1) - B),(V9 - (V * uB))]| `2 is V28() real ext-real Element of REAL
|[((X `1) - B),(V9 - (V * uB))]| `1 is V28() real ext-real Element of REAL
|.|[((X `1) - B),(V9 - (V * uB))]|.| is V28() real ext-real non negative Element of REAL
|.|[((X `1) - B),(V9 - (V * uB))]|.| ^2 is V28() real ext-real Element of REAL
|.|[((X `1) - B),(V9 - (V * uB))]|.| * |.|[((X `1) - B),(V9 - (V * uB))]|.| is V28() real ext-real non negative set
|[B,(V * uB)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
X - |[B,(V * uB)]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(X,|[B,(V * uB)]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(X - |[B,(V * uB)]|).| is V28() real ext-real non negative Element of REAL
|.(X - |[B,(V * uB)]|).| ^2 is V28() real ext-real Element of REAL
|.(X - |[B,(V * uB)]|).| * |.(X - |[B,(V * uB)]|).| is V28() real ext-real non negative set
|[B,(V * x)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
X - |[B,(V * x)]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(X,|[B,(V * x)]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(X - |[B,(V * x)]|).| is V28() real ext-real non negative Element of REAL
(V * x) - |.(X - |[B,(V * x)]|).| is V28() real ext-real Element of REAL
((V * x) - |.(X - |[B,(V * x)]|).|) + |.(X - |[B,(V * x)]|).| is V28() real ext-real Element of REAL
|.(X - |[B,(V * uB)]|).| - (V * uB) is V28() real ext-real Element of REAL
min (((V * x) - |.(X - |[B,(V * x)]|).|),(|.(X - |[B,(V * uB)]|).| - (V * uB))) is V28() real ext-real set
|[B,(V * x)]| - X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[B,(V * x)]|,X) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[B,(V * x)]| - X).| is V28() real ext-real non negative Element of REAL
|[((X `1) - B),(V9 - (V * x))]| `1 is V28() real ext-real Element of REAL
|.|[((X `1) - B),(V9 - (V * x))]|.| is V28() real ext-real non negative Element of REAL
|.|[((X `1) - B),(V9 - (V * x))]|.| ^2 is V28() real ext-real Element of REAL
|.|[((X `1) - B),(V9 - (V * x))]|.| * |.|[((X `1) - B),(V9 - (V * x))]|.| is V28() real ext-real non negative set
|.(X - |[B,(V * x)]|).| ^2 is V28() real ext-real Element of REAL
|.(X - |[B,(V * x)]|).| * |.(X - |[B,(V * x)]|).| is V28() real ext-real non negative set
x9 is non empty V28() real ext-real positive non negative set
V * x9 is non empty V28() real ext-real positive non negative set
f is non empty V28() real ext-real positive non negative set
Ball (X,f) is Element of bool the carrier of (TOP-REAL 2)
|.(|[B,(V * x)]| - X).| + f is non empty V28() real ext-real positive non negative Element of REAL
|[(X `1),0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
X - |[(X `1),0]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(X,|[(X `1),0]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(X - |[(X `1),0]|).| is V28() real ext-real non negative Element of REAL
(X `1) - (X `1) is V28() real ext-real Element of REAL
V9 - 0 is V28() real ext-real Element of REAL
|[((X `1) - (X `1)),(V9 - 0)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|.|[((X `1) - (X `1)),(V9 - 0)]|.| is V28() real ext-real non negative Element of REAL
abs V9 is V28() real ext-real Element of REAL
|[B,(V * x)]| - |[(X `1),0]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[B,(V * x)]|,|[(X `1),0]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[B,(V * x)]| - |[(X `1),0]|).| is V28() real ext-real non negative Element of REAL
|.(|[B,(V * x)]| - X).| + V9 is non empty V28() real ext-real positive non negative Element of REAL
|[(X `1),0]| - |[B,(V * x)]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[(X `1),0]|,|[B,(V * x)]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[(X `1),0]| - |[B,(V * x)]|).| is V28() real ext-real non negative Element of REAL
Ball (|[B,(V * x)]|,(V * x)) is Element of bool the carrier of (TOP-REAL 2)
Ball (|[B,(V * x)]|,(V * x9)) is Element of bool the carrier of (TOP-REAL 2)
f is set
u is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
u - X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(u,X) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(u - X).| is V28() real ext-real non negative Element of REAL
f + |.(X - |[B,(V * x)]|).| is non empty V28() real ext-real positive non negative Element of REAL
|.(u - X).| + |.(X - |[B,(V * x)]|).| is V28() real ext-real non negative Element of REAL
u - |[B,(V * x)]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(u,|[B,(V * x)]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(u - |[B,(V * x)]|).| is V28() real ext-real non negative Element of REAL
(B,V) . u is V28() real ext-real set
X - u is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(X,u) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(X - u).| is V28() real ext-real non negative Element of REAL
u - |[B,(V * uB)]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(u,|[B,(V * uB)]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(u - |[B,(V * uB)]|).| is V28() real ext-real non negative Element of REAL
|.(X - u).| + |.(u - |[B,(V * uB)]|).| is V28() real ext-real non negative Element of REAL
|.(X - |[B,(V * uB)]|).| - |.(u - X).| is V28() real ext-real Element of REAL
u `1 is V28() real ext-real Element of REAL
u `2 is V28() real ext-real Element of REAL
|[(u `1),(u `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
X is V28() real ext-real set
uB is non empty V28() real ext-real positive non negative set
B is non empty V28() real ext-real positive non negative set
uB * B is non empty V28() real ext-real positive non negative set
|[X,(uB * B)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[X,(uB * B)]|,(uB * B)) is Element of bool the carrier of (TOP-REAL 2)
(X,uB) is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
].0,B.[ is V169() V170() V171() V267() V268() V272() Element of bool REAL
(X,uB) " ].0,B.[ is Element of bool the carrier of ()
x is set
V is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
V9 is Element of the carrier of ()
V `1 is V28() real ext-real Element of REAL
V `2 is V28() real ext-real Element of REAL
|[(V `1),(V `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
(X,uB) . V is V28() real ext-real set
|[X,0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
(X,uB) . V9 is V28() real ext-real Element of the carrier of I[01]
V - |[X,(uB * B)]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(V,|[X,(uB * B)]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(V - |[X,(uB * B)]|).| is V28() real ext-real non negative Element of REAL
X is V28() real ext-real set
|[X,0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
{|[X,0]|} is functional non empty trivial finite V37() 1 -element Element of bool the carrier of (TOP-REAL 2)
uB is non empty V28() real ext-real positive non negative set
B is non empty V28() real ext-real positive non negative set
uB * B is non empty V28() real ext-real positive non negative set
|[X,(uB * B)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[X,(uB * B)]|,(uB * B)) is Element of bool the carrier of (TOP-REAL 2)
(Ball (|[X,(uB * B)]|,(uB * B))) \/ {|[X,0]|} is Element of bool the carrier of (TOP-REAL 2)
(X,uB) is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
[.0,B.[ is V169() V170() V171() V268() V272() Element of bool REAL
(X,uB) " [.0,B.[ is Element of bool the carrier of ()
x is set
V is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
V9 is Element of the carrier of ()
(X,uB) . V9 is V28() real ext-real Element of the carrier of I[01]
V - |[X,(uB * B)]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(V,|[X,(uB * B)]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(V - |[X,(uB * B)]|).| is V28() real ext-real non negative Element of REAL
(X,uB) . V is V28() real ext-real set
(X,uB) . x is V28() real ext-real set
X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
X `2 is V28() real ext-real Element of REAL
B is V28() real ext-real set
uB is V28() real ext-real set
X `1 is V28() real ext-real Element of REAL
|[(X `1),(X `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
x is non empty V28() real ext-real positive non negative set
(B,x) is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
(B,x) . X is V28() real ext-real set
x * uB is V28() real ext-real set
|[B,(x * uB)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[B,(x * uB)]|,(x * uB)) is Element of bool the carrier of (TOP-REAL 2)
|[(X `1),0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[B,0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
X - |[B,(x * uB)]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(X,|[B,(x * uB)]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(X - |[B,(x * uB)]|).| is V28() real ext-real non negative Element of REAL
|[B,x]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[B,x]|,x) is Element of bool the carrier of (TOP-REAL 2)
X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
X `2 is V28() real ext-real Element of REAL
uB is V28() real ext-real set
B is V28() real ext-real set
].uB,1.] is V169() V170() V171() V267() V272() Element of bool REAL
x is non empty V28() real ext-real positive non negative set
(B,x) is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
(B,x) . X is V28() real ext-real set
(B,x) " ].uB,1.] is Element of bool the carrier of ()
X `1 is V28() real ext-real Element of REAL
|[(X `1),(X `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
V9 is non empty V28() real ext-real positive non negative set
Ball (X,V9) is Element of bool the carrier of (TOP-REAL 2)
r is set
f is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
f `1 is V28() real ext-real Element of REAL
f `2 is V28() real ext-real Element of REAL
|[(f `1),(f `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
x9 is Element of bool the carrier of ()
f is Element of the carrier of ()
(B,x) . f is V28() real ext-real Element of the carrier of I[01]
|[B,0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[B,0]| `2 is V28() real ext-real Element of REAL
(B,x) . f is V28() real ext-real set
x * uB is V28() real ext-real set
|[B,(x * uB)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
X - |[B,(x * uB)]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(X,|[B,(x * uB)]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(X - |[B,(x * uB)]|).| is V28() real ext-real non negative Element of REAL
|.(X - |[B,(x * uB)]|).| - (x * uB) is V28() real ext-real Element of REAL
min ((|.(X - |[B,(x * uB)]|).| - (x * uB)),(X `2)) is V28() real ext-real set
V9 is non empty V28() real ext-real positive non negative set
r is non empty V28() real ext-real positive non negative set
Ball (X,r) is Element of bool the carrier of (TOP-REAL 2)
f is set
f is Element of bool the carrier of ()
u is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
u `1 is V28() real ext-real Element of REAL
u `2 is V28() real ext-real Element of REAL
|[(u `1),(u `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
(x * uB) + r is V28() real ext-real set
(|.(X - |[B,(x * uB)]|).| - (x * uB)) + (x * uB) is V28() real ext-real Element of REAL
b is Element of the carrier of ()
(B,x) . b is V28() real ext-real Element of the carrier of I[01]
u - |[B,(x * uB)]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(u,|[B,(x * uB)]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(u - |[B,(x * uB)]|).| is V28() real ext-real non negative Element of REAL
|[B,(x * uB)]| - u is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[B,(x * uB)]|,u) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[B,(x * uB)]| - u).| is V28() real ext-real non negative Element of REAL
|[B,(x * uB)]| - X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[B,(x * uB)]|,X) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[B,(x * uB)]| - X).| is V28() real ext-real non negative Element of REAL
u - X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(u,X) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(u - X).| is V28() real ext-real non negative Element of REAL
|.(|[B,(x * uB)]| - u).| + |.(u - X).| is V28() real ext-real non negative Element of REAL
X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
X `2 is V28() real ext-real Element of REAL
X `1 is V28() real ext-real Element of REAL
{X} is functional non empty trivial finite V37() 1 -element Element of bool the carrier of (TOP-REAL 2)
B is V28() real ext-real set
uB is non empty V28() real ext-real positive non negative set
(B,uB) is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
(B,uB) . X is V28() real ext-real set
(B,uB) " {1} is Element of bool the carrier of ()
|[B,uB]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
X - |[B,uB]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(X,|[B,uB]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(X - |[B,uB]|).| is V28() real ext-real non negative Element of REAL
|.(X - |[B,uB]|).| - uB is V28() real ext-real Element of REAL
(|.(X - |[B,uB]|).| - uB) / 2 is V28() real ext-real Element of REAL
|[(X `1),(X `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
(X `1) - B is V28() real ext-real Element of REAL
((X `1) - B) ^2 is V28() real ext-real Element of REAL
((X `1) - B) * ((X `1) - B) is V28() real ext-real set
0 - uB is V28() real ext-real Element of REAL
(0 - uB) ^2 is V28() real ext-real Element of REAL
(0 - uB) * (0 - uB) is V28() real ext-real set
(((X `1) - B) ^2) + ((0 - uB) ^2) is V28() real ext-real Element of REAL
0 + ((0 - uB) ^2) is V28() real ext-real Element of REAL
(X `2) - uB is V28() real ext-real Element of REAL
|[((X `1) - B),((X `2) - uB)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|.|[((X `1) - B),((X `2) - uB)]|.| is V28() real ext-real non negative Element of REAL
|.|[((X `1) - B),((X `2) - uB)]|.| ^2 is V28() real ext-real Element of REAL
|.|[((X `1) - B),((X `2) - uB)]|.| * |.|[((X `1) - B),((X `2) - uB)]|.| is V28() real ext-real non negative set
uB ^2 is V28() real ext-real set
uB * uB is non empty V28() real ext-real positive non negative set
|.(X - |[B,uB]|).| ^2 is V28() real ext-real Element of REAL
|.(X - |[B,uB]|).| * |.(X - |[B,uB]|).| is V28() real ext-real non negative set
V9 is non empty V28() real ext-real positive non negative set
|[(X `1),V9]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[(X `1),V9]|,V9) is Element of bool the carrier of (TOP-REAL 2)
(Ball (|[(X `1),V9]|,V9)) \/ {X} is Element of bool the carrier of (TOP-REAL 2)
x9 is set
r is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
r `1 is V28() real ext-real Element of REAL
r `2 is V28() real ext-real Element of REAL
|[(r `1),(r `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
r - |[(X `1),V9]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(r,|[(X `1),V9]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(r - |[(X `1),V9]|).| is V28() real ext-real non negative Element of REAL
|[(X `1),V9]| - X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[(X `1),V9]|,X) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[(X `1),V9]| - X).| is V28() real ext-real non negative Element of REAL
V9 + |.(|[(X `1),V9]| - X).| is non empty V28() real ext-real positive non negative Element of REAL
|.(r - |[(X `1),V9]|).| + |.(|[(X `1),V9]| - X).| is V28() real ext-real non negative Element of REAL
|.V9.| is V28() real ext-real Element of REAL
r - X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(r,X) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(r - X).| is V28() real ext-real non negative Element of REAL
|[B,uB]| - X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[B,uB]|,X) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[B,uB]| - X).| is V28() real ext-real non negative Element of REAL
|[B,uB]| - r is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[B,uB]|,r) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[B,uB]| - r).| is V28() real ext-real non negative Element of REAL
|.(r - X).| + |.(|[B,uB]| - r).| is V28() real ext-real non negative Element of REAL
|[0,V9]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|.|[0,V9]|.| is V28() real ext-real non negative Element of REAL
(X `1) - (X `1) is V28() real ext-real Element of REAL
V9 - 0 is V28() real ext-real Element of REAL
|[((X `1) - (X `1)),(V9 - 0)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
V9 + V9 is non empty V28() real ext-real positive non negative set
(|.(X - |[B,uB]|).| - uB) + |.(|[B,uB]| - r).| is V28() real ext-real Element of REAL
|.(|[B,uB]| - X).| - (|.(X - |[B,uB]|).| - uB) is V28() real ext-real Element of REAL
r - |[B,uB]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(r,|[B,uB]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(r - |[B,uB]|).| is V28() real ext-real non negative Element of REAL
Ball (|[B,uB]|,uB) is Element of bool the carrier of (TOP-REAL 2)
f is Element of the carrier of ()
(B,uB) . f is V28() real ext-real Element of the carrier of I[01]
X is non empty TopSpace-like TopStruct
the carrier of X is non empty set
bool the carrier of X is non empty set
bool (bool the carrier of X) is non empty set
B is TopSpace-like SubSpace of X
the carrier of B is set
[#] B is non proper open closed dense Element of bool the carrier of B
bool the carrier of B is non empty set
bool (bool the carrier of B) is non empty set
uB is open quasi_basis Element of bool (bool the carrier of X)
{ (b<sub>1</sub> /\ ([#] B)) where b<sub>1</sub> is Element of bool the carrier of X : ( b<sub>1</sub> in uB & not b<sub>1</sub> misses [#] B ) } is set
bool the carrier of B is non empty Element of bool (bool the carrier of B)
V is set
V9 is Element of bool the carrier of X
V9 /\ ([#] B) is Element of bool the carrier of B
V9 is Element of bool the carrier of B
x9 is Element of bool the carrier of X
x9 /\ the carrier of B is Element of bool the carrier of X
r is Element of the carrier of B
f is Element of bool the carrier of X
f /\ ([#] B) is Element of bool the carrier of B
f is Element of bool the carrier of B
u is Element of bool the carrier of B
V is Element of bool (bool the carrier of B)
the topology of B is non empty Element of bool (bool the carrier of B)
V9 is set
the topology of X is non empty Element of bool (bool the carrier of X)
x9 is Element of bool the carrier of X
x9 /\ ([#] B) is Element of bool the carrier of B
{ ].b<sub>1</sub>,b<sub>2</sub>.[ where b<sub>1</sub>, b<sub>2</sub> is V28() real ext-real Element of REAL : not b<sub>2</sub> <= b<sub>1</sub> } is set
bool (bool the carrier of R^1) is non empty set
bool REAL is non empty non trivial non finite Element of bool (bool REAL)
B is set
uB is V28() real ext-real Element of REAL
x is V28() real ext-real Element of REAL
].uB,x.[ is V169() V170() V171() V267() V268() V272() Element of bool REAL
uB is V169() V170() V171() Element of bool the carrier of R^1
x is V28() real ext-real Element of the carrier of R^1
V is V28() real ext-real Element of REAL
V9 is V28() real ext-real Element of REAL
V - V9 is V28() real ext-real Element of REAL
V + V9 is V28() real ext-real Element of REAL
].(V - V9),(V + V9).[ is V169() V170() V171() V267() V268() V272() Element of bool REAL
x9 is V169() V170() V171() Element of bool the carrier of R^1
r is V169() V170() V171() Element of bool the carrier of R^1
B is Element of bool (bool the carrier of R^1)
the topology of R^1 is non empty Element of bool (bool the carrier of R^1)
uB is set
x is V28() real ext-real Element of REAL
V is V28() real ext-real Element of REAL
].x,V.[ is V169() V170() V171() V267() V268() V272() Element of bool REAL
X is TopSpace-like TopStruct
the carrier of X is set
bool the carrier of X is non empty set
bool (bool the carrier of X) is non empty set
B is Element of bool the carrier of X
uB is Element of bool the carrier of X
B \/ uB is Element of bool the carrier of X
{(B \/ uB)} is non empty trivial finite 1 -element Element of bool (bool the carrier of X)
x is set
x \/ {(B \/ uB)} is set
V9 is Element of bool the carrier of X
x9 is Element of the carrier of X
r is Element of bool the carrier of X
V is Element of bool (bool the carrier of X)
the topology of X is non empty Element of bool (bool the carrier of X)
{ [.0,b<sub>1</sub>.[ where b<sub>1</sub> is V28() real ext-real Element of REAL : ( not b<sub>1</sub> <= 0 & b<sub>1</sub> <= 1 ) } is set
{ ].b<sub>1</sub>,1.] where b<sub>1</sub> is V28() real ext-real Element of REAL : ( 0 <= b<sub>1</sub> & not 1 <= b<sub>1</sub> ) } is set
{ [.0,b<sub>1</sub>.[ where b<sub>1</sub> is V28() real ext-real Element of REAL : ( not b<sub>1</sub> <= 0 & b<sub>1</sub> <= 1 ) } \/ { ].b<sub>1</sub>,1.] where b<sub>1</sub> is V28() real ext-real Element of REAL : ( 0 <= b<sub>1</sub> & not 1 <= b<sub>1</sub> ) } is set
{ ].b<sub>1</sub>,b<sub>2</sub>.[ where b<sub>1</sub>, b<sub>2</sub> is V28() real ext-real Element of REAL : ( 0 <= b<sub>1</sub> & not b<sub>2</sub> <= b<sub>1</sub> & b<sub>2</sub> <= 1 ) } is set
( { [.0,b<sub>1</sub>.[ where b<sub>1</sub> is V28() real ext-real Element of REAL : ( not b<sub>1</sub> <= 0 & b<sub>1</sub> <= 1 ) } \/ { ].b<sub>1</sub>,1.] where b<sub>1</sub> is V28() real ext-real Element of REAL : ( 0 <= b<sub>1</sub> & not 1 <= b<sub>1</sub> ) } ) \/ { ].b<sub>1</sub>,b<sub>2</sub>.[ where b<sub>1</sub>, b<sub>2</sub> is V28() real ext-real Element of REAL : ( 0 <= b<sub>1</sub> & not b<sub>2</sub> <= b<sub>1</sub> & b<sub>2</sub> <= 1 ) } is set
bool the carrier of I[01] is non empty set
bool (bool the carrier of I[01]) is non empty set
2 / 3 is non empty V28() real ext-real positive non negative rational Element of REAL
[.0,(2 / 3).[ is V169() V170() V171() V268() V272() Element of bool REAL
1 / 3 is non empty V28() real ext-real positive non negative rational Element of REAL
].(1 / 3),1.] is V169() V170() V171() V267() V272() Element of bool REAL
[#] I[01] is non empty non proper open closed dense non boundary V169() V170() V171() Element of bool the carrier of I[01]
uB is open quasi_basis Element of bool (bool the carrier of R^1)
{ (b<sub>1</sub> /\ ([#] I[01])) where b<sub>1</sub> is V169() V170() V171() Element of bool the carrier of R^1 : ( b<sub>1</sub> in uB & not b<sub>1</sub> misses [#] I[01] ) } is set
{([#] I[01])} is non empty trivial finite 1 -element Element of bool (bool the carrier of I[01])
(( { [.0,b<sub>1</sub>.[ where b<sub>1</sub> is V28() real ext-real Element of REAL : ( not b<sub>1</sub> <= 0 & b<sub>1</sub> <= 1 ) } \/ { ].b<sub>1</sub>,1.] where b<sub>1</sub> is V28() real ext-real Element of REAL : ( 0 <= b<sub>1</sub> & not 1 <= b<sub>1</sub> ) } ) \/ { ].b<sub>1</sub>,b<sub>2</sub>.[ where b<sub>1</sub>, b<sub>2</sub> is V28() real ext-real Element of REAL : ( 0 <= b<sub>1</sub> & not b<sub>2</sub> <= b<sub>1</sub> & b<sub>2</sub> <= 1 ) } ) \/ {([#] I[01])} is set
].(- 1),2.[ is V169() V170() V171() V267() V268() V272() Element of bool REAL
u is set
b is V169() V170() V171() Element of bool the carrier of R^1
b /\ ([#] I[01]) is V169() V170() V171() Element of bool the carrier of I[01]
c is set
A is V28() real ext-real Element of REAL
b is V28() real ext-real Element of REAL
].A,b.[ is V169() V170() V171() V267() V268() V272() Element of bool REAL
x is V28() real ext-real Element of REAL
[.0,b.[ is V169() V170() V171() V268() V272() Element of bool REAL
{ ].b<sub>1</sub>,b<sub>2</sub>.[ where b<sub>1</sub>, b<sub>2</sub> is V28() real ext-real Element of REAL : ( 0 <= b<sub>1</sub> & not b<sub>2</sub> <= b<sub>1</sub> & b<sub>2</sub> <= 1 ) } \/ {([#] I[01])} is set
( { [.0,b<sub>1</sub>.[ where b<sub>1</sub> is V28() real ext-real Element of REAL : ( not b<sub>1</sub> <= 0 & b<sub>1</sub> <= 1 ) } \/ { ].b<sub>1</sub>,1.] where b<sub>1</sub> is V28() real ext-real Element of REAL : ( 0 <= b<sub>1</sub> & not 1 <= b<sub>1</sub> ) } ) \/ ( { ].b<sub>1</sub>,b<sub>2</sub>.[ where b<sub>1</sub>, b<sub>2</sub> is V28() real ext-real Element of REAL : ( 0 <= b<sub>1</sub> & not b<sub>2</sub> <= b<sub>1</sub> & b<sub>2</sub> <= 1 ) } \/ {([#] I[01])}) is set
].A,1.] is V169() V170() V171() V267() V272() Element of bool REAL
{ ].b<sub>1</sub>,b<sub>2</sub>.[ where b<sub>1</sub>, b<sub>2</sub> is V28() real ext-real Element of REAL : ( 0 <= b<sub>1</sub> & not b<sub>2</sub> <= b<sub>1</sub> & b<sub>2</sub> <= 1 ) } \/ {([#] I[01])} is set
( { [.0,b<sub>1</sub>.[ where b<sub>1</sub> is V28() real ext-real Element of REAL : ( not b<sub>1</sub> <= 0 & b<sub>1</sub> <= 1 ) } \/ { ].b<sub>1</sub>,1.] where b<sub>1</sub> is V28() real ext-real Element of REAL : ( 0 <= b<sub>1</sub> & not 1 <= b<sub>1</sub> ) } ) \/ ( { ].b<sub>1</sub>,b<sub>2</sub>.[ where b<sub>1</sub>, b<sub>2</sub> is V28() real ext-real Element of REAL : ( 0 <= b<sub>1</sub> & not b<sub>2</sub> <= b<sub>1</sub> & b<sub>2</sub> <= 1 ) } \/ {([#] I[01])}) is set
u is set
f is V169() V170() V171() Element of bool the carrier of R^1
f /\ ([#] I[01]) is V169() V170() V171() Element of bool the carrier of I[01]
b is V28() real ext-real Element of REAL
[.0,b.[ is V169() V170() V171() V268() V272() Element of bool REAL
].(- 1),b.[ is V169() V170() V171() V267() V268() V272() Element of bool REAL
c is V169() V170() V171() Element of bool the carrier of R^1
c /\ [.0,1.] is V169() V170() V171() Element of bool REAL
b is V28() real ext-real Element of REAL
].b,1.] is V169() V170() V171() V267() V272() Element of bool REAL
].b,2.[ is V169() V170() V171() V267() V268() V272() Element of bool REAL
c is V169() V170() V171() Element of bool the carrier of R^1
c /\ [.0,1.] is V169() V170() V171() Element of bool REAL
b is V28() real ext-real Element of REAL
c is V28() real ext-real Element of REAL
].b,c.[ is V169() V170() V171() V267() V268() V272() Element of bool REAL
A is V169() V170() V171() Element of bool the carrier of R^1
c + c is V28() real ext-real Element of REAL
b + c is V28() real ext-real Element of REAL
(c + c) / 2 is V28() real ext-real Element of REAL
(b + c) / 2 is V28() real ext-real Element of REAL
b + b is V28() real ext-real Element of REAL
(b + b) / 2 is V28() real ext-real Element of REAL
A /\ [.0,1.] is V169() V170() V171() Element of bool REAL
X is V169() V170() V171() Element of bool the carrier of I[01]
B is V169() V170() V171() Element of bool the carrier of I[01]
X \/ B is V169() V170() V171() Element of bool the carrier of I[01]
V is non empty TopSpace-like TopStruct
the carrier of V is non empty set
[: the carrier of V, the carrier of I[01]:] is Relation-like non empty V159() V160() V161() set
bool [: the carrier of V, the carrier of I[01]:] is non empty set
V9 is Relation-like the carrier of V -defined the carrier of I[01] -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of V, the carrier of I[01]:]
x9 is V28() real ext-real set
r is V28() real ext-real set
f is V28() real ext-real Element of REAL
].f,1.] is V169() V170() V171() V267() V272() Element of bool REAL
x is open quasi_basis Element of bool (bool the carrier of I[01])
f is V28() real ext-real Element of REAL
[.0,f.[ is V169() V170() V171() V268() V272() Element of bool REAL
[.0,r.[ is V169() V170() V171() V268() V272() Element of bool REAL
V9 " [.0,r.[ is Element of bool the carrier of V
bool the carrier of V is non empty set
].x9,1.] is V169() V170() V171() V267() V272() Element of bool REAL
V9 " ].x9,1.] is Element of bool the carrier of V
x9 is V169() V170() V171() Element of bool the carrier of I[01]
V9 " x9 is Element of bool the carrier of V
r is V28() real ext-real Element of REAL
[.0,r.[ is V169() V170() V171() V268() V272() Element of bool REAL
V9 " x9 is Element of bool the carrier of V
r is V28() real ext-real Element of REAL
].r,1.] is V169() V170() V171() V267() V272() Element of bool REAL
r is V28() real ext-real Element of REAL
f is V28() real ext-real Element of REAL
].r,f.[ is V169() V170() V171() V267() V268() V272() Element of bool REAL
[.0,f.[ is V169() V170() V171() V268() V272() Element of bool REAL
V9 " [.0,f.[ is Element of bool the carrier of V
].r,1.] is V169() V170() V171() V267() V272() Element of bool REAL
V9 " ].r,1.] is Element of bool the carrier of V
[.0,f.[ /\ ].r,1.] is V169() V170() V171() V272() Element of bool REAL
V9 " x9 is Element of bool the carrier of V
f is open Element of bool the carrier of V
u is open Element of bool the carrier of V
f /\ u is open Element of bool the carrier of V
X is V28() real ext-real set
B is non empty V28() real ext-real positive non negative set
(X,B) is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
x is Relation-like the carrier of () -defined Function-like total Neighborhood_System of ()
dom x is Element of bool the carrier of ()
V is V28() real ext-real set
V9 is V28() real ext-real set
x9 is Element of the carrier of ()
[.0,V9.[ is V169() V170() V171() V268() V272() Element of bool REAL
(X,B) " [.0,V9.[ is Element of bool the carrier of ()
(X,B) . x9 is V28() real ext-real Element of the carrier of I[01]
r is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
r `1 is V28() real ext-real Element of REAL
r `2 is V28() real ext-real Element of REAL
|[(r `1),(r `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
B * V9 is V28() real ext-real set
f is non empty V28() real ext-real positive non negative set
|[X,f]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[X,f]|,f) is Element of bool the carrier of (TOP-REAL 2)
|[X,0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
{|[X,0]|} is functional non empty trivial finite V37() 1 -element Element of bool the carrier of (TOP-REAL 2)
(Ball (|[X,f]|,f)) \/ {|[X,0]|} is Element of bool the carrier of (TOP-REAL 2)
f is Element of bool the carrier of ()
u is Element of bool the carrier of ()
{ ((Ball (|[X,b<sub>1</sub>]|,b<sub>1</sub>)) \/ {|[X,0]|}) where b<sub>1</sub> is V28() real ext-real Element of REAL : not b<sub>1</sub> <= 0 } is set
x . |[X,0]| is set
Union x is open quasi_basis Element of bool (bool the carrier of ())
bool (bool the carrier of ()) is non empty set
B * V9 is V28() real ext-real set
f is non empty V28() real ext-real positive non negative set
|[X,f]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[X,f]|,f) is Element of bool the carrier of (TOP-REAL 2)
f is Element of bool the carrier of ()
u is Element of bool the carrier of ()
u /\ () is Element of bool the carrier of (TOP-REAL 2)
{ ((Ball (|[X,f]|,b<sub>1</sub>)) /\ ()) where b<sub>1</sub> is V28() real ext-real Element of REAL : not b<sub>1</sub> <= 0 } is set
x . |[X,f]| is set
Union x is open quasi_basis Element of bool (bool the carrier of ())
bool (bool the carrier of ()) is non empty set
].0,V9.[ is V169() V170() V171() V267() V268() V272() Element of bool REAL
(X,B) " ].0,V9.[ is Element of bool the carrier of ()
|[X,(B * V9)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[X,(B * V9)]|,(B * V9)) is Element of bool the carrier of (TOP-REAL 2)
Union x is open quasi_basis Element of bool (bool the carrier of ())
bool (bool the carrier of ()) is non empty set
Union x is open quasi_basis Element of bool (bool the carrier of ())
bool (bool the carrier of ()) is non empty set
x9 is Element of the carrier of ()
].V,1.] is V169() V170() V171() V267() V272() Element of bool REAL
(X,B) " ].V,1.] is Element of bool the carrier of ()
(X,B) . x9 is V28() real ext-real Element of the carrier of I[01]
r is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
r `1 is V28() real ext-real Element of REAL
r `2 is V28() real ext-real Element of REAL
|[(r `1),(r `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
f is non empty V28() real ext-real positive non negative set
Ball (r,f) is Element of bool the carrier of (TOP-REAL 2)
f is Element of bool the carrier of ()
f /\ () is Element of bool the carrier of (TOP-REAL 2)
{ ((Ball (|[(r `1),(r `2)]|,b<sub>1</sub>)) /\ ()) where b<sub>1</sub> is V28() real ext-real Element of REAL : not b<sub>1</sub> <= 0 } is set
x . |[(r `1),(r `2)]| is set
u is Element of bool the carrier of ()
{r} is functional non empty trivial finite V37() 1 -element Element of bool the carrier of (TOP-REAL 2)
(X,B) " {1} is Element of bool the carrier of ()
f is non empty V28() real ext-real positive non negative set
|[(r `1),f]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[(r `1),f]|,f) is Element of bool the carrier of (TOP-REAL 2)
(Ball (|[(r `1),f]|,f)) \/ {r} is Element of bool the carrier of (TOP-REAL 2)
f is Element of bool the carrier of ()
|[(r `1),0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
{|[(r `1),0]|} is functional non empty trivial finite V37() 1 -element Element of bool the carrier of (TOP-REAL 2)
{ ((Ball (|[(r `1),b<sub>1</sub>]|,b<sub>1</sub>)) \/ {|[(r `1),0]|}) where b₁ is V28() real ext-real Element of REAL : not b₁ <= 0 } is set
x . |[(r `1),(r `2)]| is set
u is Element of bool the carrier of ()
].V,1.[ is V169() V170() V171() V267() V268() V272() Element of bool REAL
(X,B) " ].V,1.[ is Element of bool the carrier of ()
f is non empty V28() real ext-real positive non negative set
Ball (r,f) is Element of bool the carrier of (TOP-REAL 2)
f is Element of bool the carrier of ()
f /\ () is Element of bool the carrier of (TOP-REAL 2)
{ ((Ball (|[(r `1),(r `2)]|,b₁)) /\ ()) where b₁ is V28() real ext-real Element of REAL : not b₁ <= 0 } is set
x . |[(r `1),(r `2)]| is set
u is Element of bool the carrier of ()
X is Element of bool the carrier of ()
X ` is Element of bool the carrier of ()
the carrier of () \ X is set
B is V28() real ext-real Element of REAL
|[B,0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
{|[B,0]|} is functional non empty trivial finite V37() 1 -element Element of bool the carrier of (TOP-REAL 2)
X \ {|[B,0]|} is Element of bool the carrier of ()
uB is non empty V28() real ext-real positive non negative set
|[B,uB]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[B,uB]|,uB) is Element of bool the carrier of (TOP-REAL 2)
(Ball (|[B,uB]|,uB)) \/ {|[B,0]|} is Element of bool the carrier of (TOP-REAL 2)
2 * uB is non empty V28() real ext-real positive non negative Element of REAL
(B,uB) is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
x is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
x . |[B,0]| is V28() real ext-real set
V is V28() real ext-real set
V9 is V28() real ext-real set
|[V,V9]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
x . |[V,V9]| is V28() real ext-real set
|[B,0]| - |[V,V9]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[B,0]|,|[V,V9]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[B,0]| - |[V,V9]|).| is V28() real ext-real non negative Element of REAL
|.(|[B,0]| - |[V,V9]|).| ^2 is V28() real ext-real Element of REAL
|.(|[B,0]| - |[V,V9]|).| * |.(|[B,0]| - |[V,V9]|).| is V28() real ext-real non negative set
(2 * uB) * V9 is V28() real ext-real Element of REAL
(|.(|[B,0]| - |[V,V9]|).| ^2) / ((2 * uB) * V9) is V28() real ext-real Element of REAL
X is V28() real ext-real set
B is V28() real ext-real set
|[X,B]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
uB is non empty V28() real ext-real positive non negative set
Ball (|[X,B]|,uB) is Element of bool the carrier of (TOP-REAL 2)
x is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[X,B]| - x is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[X,B]|,x) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[X,B]| - x).| is V28() real ext-real non negative Element of REAL
|.(|[X,B]| - x).| / uB is V28() real ext-real non negative Element of REAL
x - |[X,B]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(x,|[X,B]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(x - |[X,B]|).| is V28() real ext-real non negative Element of REAL
uB / uB is non empty V28() real ext-real positive non negative Element of COMPLEX
x is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
V is V28() real ext-real set
V9 is V28() real ext-real non negative set
|[V,V9]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
x . |[V,V9]| is V28() real ext-real set
|[X,B]| - |[V,V9]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[X,B]|,|[V,V9]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[X,B]| - |[V,V9]|).| is V28() real ext-real non negative Element of REAL
|.(|[X,B]| - |[V,V9]|).| / uB is V28() real ext-real non negative Element of REAL
x is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
V is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
V9 is Element of the carrier of ()
x9 is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
x9 `1 is V28() real ext-real Element of REAL
x9 `2 is V28() real ext-real Element of REAL
|[(x9 `1),(x9 `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
r is V28() real ext-real non negative set
|[(x9 `1),r]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
x . V9 is V28() real ext-real Element of the carrier of I[01]
V . V9 is V28() real ext-real Element of the carrier of I[01]
r is V28() real ext-real non negative set
|[(x9 `1),r]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
x . V9 is V28() real ext-real Element of the carrier of I[01]
|[X,B]| - x9 is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[X,B]|,x9) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[X,B]| - x9).| is V28() real ext-real non negative Element of REAL
|.(|[X,B]| - x9).| / uB is V28() real ext-real non negative Element of REAL
V . V9 is V28() real ext-real Element of the carrier of I[01]
r is V28() real ext-real non negative set
|[(x9 `1),r]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
X `2 is V28() real ext-real Element of REAL
B is V28() real ext-real set
uB is V28() real ext-real non negative set
|[B,uB]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
x is non empty V28() real ext-real positive non negative set
(B,uB,x) is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
(B,uB,x) . X is V28() real ext-real set
X `1 is V28() real ext-real Element of REAL
|[(X `1),(X `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[B,uB]|,x) is Element of bool the carrier of (TOP-REAL 2)
|[B,uB]| - X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[B,uB]|,X) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[B,uB]| - X).| is V28() real ext-real non negative Element of REAL
|.(|[B,uB]| - X).| / x is V28() real ext-real non negative Element of REAL
0 * x is V28() real ext-real Element of REAL
0 / x is V28() real ext-real Element of REAL
X is V28() real ext-real set
B is V28() real ext-real non negative set
|[X,B]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
x is non empty V28() real ext-real positive non negative set
uB is non empty V28() real ext-real positive non negative set
(X,B,uB) is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
[.0,x.[ is V169() V170() V171() V268() V272() Element of bool REAL
(X,B,uB) " [.0,x.[ is Element of bool the carrier of ()
uB * x is non empty V28() real ext-real positive non negative set
Ball (|[X,B]|,(uB * x)) is Element of bool the carrier of (TOP-REAL 2)
(Ball (|[X,B]|,(uB * x))) /\ () is Element of bool the carrier of (TOP-REAL 2)
uB * 1 is non empty V28() real ext-real positive non negative Element of REAL
Ball (|[X,B]|,uB) is Element of bool the carrier of (TOP-REAL 2)
V9 is set
x9 is Element of the carrier of ()
(X,B,uB) . x9 is V28() real ext-real Element of the carrier of I[01]
r is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
r `1 is V28() real ext-real Element of REAL
r `2 is V28() real ext-real Element of REAL
|[(r `1),(r `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[X,B]| - r is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[X,B]|,r) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[X,B]| - r).| is V28() real ext-real non negative Element of REAL
|.(|[X,B]| - r).| / uB is V28() real ext-real non negative Element of REAL
r - |[X,B]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(r,|[X,B]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(r - |[X,B]|).| is V28() real ext-real non negative Element of REAL
V9 is set
x9 is Element of the carrier of ()
r is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
r - |[X,B]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(r,|[X,B]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(r - |[X,B]|).| is V28() real ext-real non negative Element of REAL
|[X,B]| - r is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[X,B]|,r) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[X,B]| - r).| is V28() real ext-real non negative Element of REAL
r `1 is V28() real ext-real Element of REAL
r `2 is V28() real ext-real Element of REAL
|[(r `1),(r `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
(X,B,uB) . x9 is V28() real ext-real Element of the carrier of I[01]
|.(|[X,B]| - r).| / uB is V28() real ext-real non negative Element of REAL
uB * ((X,B,uB) . x9) is V28() real ext-real set
X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
X `2 is V28() real ext-real Element of REAL
B is V28() real ext-real set
uB is V28() real ext-real non negative set
].uB,1.] is V169() V170() V171() V267() V272() Element of bool REAL
x is non empty V28() real ext-real positive non negative set
V is non empty V28() real ext-real positive non negative set
(B,x,V) is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
(B,x,V) . X is V28() real ext-real set
|[B,x]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[B,x]| - X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[B,x]|,X) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[B,x]| - X).| is V28() real ext-real non negative Element of REAL
V * uB is V28() real ext-real non negative set
|.(|[B,x]| - X).| - (V * uB) is V28() real ext-real Element of REAL
Ball (X,(|.(|[B,x]| - X).| - (V * uB))) is Element of bool the carrier of (TOP-REAL 2)
(Ball (X,(|.(|[B,x]| - X).| - (V * uB)))) /\ () is Element of bool the carrier of (TOP-REAL 2)
(B,x,V) " ].uB,1.] is Element of bool the carrier of ()
X `1 is V28() real ext-real Element of REAL
|[(X `1),(X `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
X - |[B,x]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(X,|[B,x]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(X - |[B,x]|).| is V28() real ext-real non negative Element of REAL
Ball (|[B,x]|,V) is Element of bool the carrier of (TOP-REAL 2)
|.(|[B,x]| - X).| / V is V28() real ext-real non negative Element of REAL
V / V is non empty V28() real ext-real positive non negative Element of COMPLEX
Ball (|[B,x]|,V) is Element of bool the carrier of (TOP-REAL 2)
|.(|[B,x]| - X).| / V is V28() real ext-real non negative Element of REAL
V * 1 is non empty V28() real ext-real positive non negative Element of REAL
r is set
f is Element of the carrier of ()
x9 is non empty V28() real ext-real positive non negative set
f is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
f - X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(f,X) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(f - X).| is V28() real ext-real non negative Element of REAL
|[B,x]| - f is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[B,x]|,f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[B,x]| - f).| is V28() real ext-real non negative Element of REAL
|.(|[B,x]| - f).| + x9 is non empty V28() real ext-real positive non negative Element of REAL
|.(|[B,x]| - f).| + |.(f - X).| is V28() real ext-real non negative Element of REAL
f `1 is V28() real ext-real Element of REAL
f `2 is V28() real ext-real Element of REAL
|[(f `1),(f `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
(|.(|[B,x]| - f).| + x9) - x9 is V28() real ext-real Element of REAL
|.(|[B,x]| - X).| - x9 is V28() real ext-real Element of REAL
Ball (|[B,x]|,V) is Element of bool the carrier of (TOP-REAL 2)
(B,x,V) . f is V28() real ext-real set
|.(|[B,x]| - f).| / V is V28() real ext-real non negative Element of REAL
(B,x,V) . f is V28() real ext-real Element of the carrier of I[01]
X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
X `2 is V28() real ext-real Element of REAL
X `1 is V28() real ext-real Element of REAL
{X} is functional non empty trivial finite V37() 1 -element Element of bool the carrier of (TOP-REAL 2)
|[(X `1),(X `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
uB is V28() real ext-real set
x is V28() real ext-real non negative set
].x,1.] is V169() V170() V171() V267() V272() Element of bool REAL
V is non empty V28() real ext-real positive non negative set
V9 is non empty V28() real ext-real positive non negative set
(uB,V,V9) is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
(uB,V,V9) . X is V28() real ext-real set
|[uB,V]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[uB,V]| - X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[uB,V]|,X) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[uB,V]| - X).| is V28() real ext-real non negative Element of REAL
V9 * x is V28() real ext-real non negative set
|.(|[uB,V]| - X).| - (V9 * x) is V28() real ext-real Element of REAL
(|.(|[uB,V]| - X).| - (V9 * x)) / 2 is V28() real ext-real Element of REAL
(uB,V,V9) " ].x,1.] is Element of bool the carrier of ()
X - |[uB,V]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(X,|[uB,V]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(X - |[uB,V]|).| is V28() real ext-real non negative Element of REAL
Ball (|[uB,V]|,V9) is Element of bool the carrier of (TOP-REAL 2)
|.(|[uB,V]| - X).| / V9 is V28() real ext-real non negative Element of REAL
V9 / V9 is non empty V28() real ext-real positive non negative Element of COMPLEX
Ball (|[uB,V]|,V9) is Element of bool the carrier of (TOP-REAL 2)
|.(|[uB,V]| - X).| / V9 is V28() real ext-real non negative Element of REAL
V9 * 1 is non empty V28() real ext-real positive non negative Element of REAL
r is non empty V28() real ext-real positive non negative set
r / 2 is non empty V28() real ext-real positive non negative Element of REAL
f is non empty V28() real ext-real positive non negative Element of REAL
|[(X `1),f]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[(X `1),f]|,f) is Element of bool the carrier of (TOP-REAL 2)
(Ball (|[(X `1),f]|,f)) \/ {X} is Element of bool the carrier of (TOP-REAL 2)
f is set
B is Element of the carrier of ()
u is Element of the carrier of ()
b is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
b `1 is V28() real ext-real Element of REAL
b `2 is V28() real ext-real Element of REAL
|[(b `1),(b `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[uB,V]|,V9) is Element of bool the carrier of (TOP-REAL 2)
(uB,V,V9) . b is V28() real ext-real set
(uB,V,V9) . u is V28() real ext-real Element of the carrier of I[01]
b - |[(X `1),f]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(b,|[(X `1),f]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(b - |[(X `1),f]|).| is V28() real ext-real non negative Element of REAL
|[(X `1),f]| - X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[(X `1),f]|,X) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[(X `1),f]| - X).| is V28() real ext-real non negative Element of REAL
f + |.(|[(X `1),f]| - X).| is non empty V28() real ext-real positive non negative Element of REAL
|.(b - |[(X `1),f]|).| + |.(|[(X `1),f]| - X).| is V28() real ext-real non negative Element of REAL
b - X is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(b,X) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(b - X).| is V28() real ext-real non negative Element of REAL
|[0,f]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|.|[0,f]|.| is V28() real ext-real non negative Element of REAL
abs f is V28() real ext-real Element of REAL
|[uB,V]| - b is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[uB,V]|,b) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[uB,V]| - b).| is V28() real ext-real non negative Element of REAL
|.(|[uB,V]| - b).| + |.(b - X).| is V28() real ext-real non negative Element of REAL
(X `1) - (X `1) is V28() real ext-real Element of REAL
f - 0 is V28() real ext-real Element of REAL
|[((X `1) - (X `1)),(f - 0)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|.|[((X `1) - (X `1)),(f - 0)]|.| is V28() real ext-real non negative Element of REAL
f + f is non empty V28() real ext-real positive non negative Element of REAL
|.(|[uB,V]| - b).| + (f + f) is non empty V28() real ext-real positive non negative Element of REAL
(|.(|[uB,V]| - b).| + (f + f)) - (f + f) is V28() real ext-real Element of REAL
|.(|[uB,V]| - X).| - (f + f) is V28() real ext-real Element of REAL
|.(|[uB,V]| - b).| / V9 is V28() real ext-real non negative Element of REAL
(uB,V,V9) . u is V28() real ext-real Element of the carrier of I[01]
X is V28() real ext-real set
B is non empty V28() real ext-real positive non negative set
uB is non empty V28() real ext-real positive non negative set
(X,B,uB) is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
V is Relation-like the carrier of () -defined Function-like total Neighborhood_System of ()
dom V is Element of bool the carrier of ()
V9 is V28() real ext-real set
x9 is V28() real ext-real set
[.0,x9.[ is V169() V170() V171() V268() V272() Element of bool REAL
(X,B,uB) " [.0,x9.[ is Element of bool the carrier of ()
|[X,B]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
uB * x9 is V28() real ext-real set
Ball (|[X,B]|,(uB * x9)) is Element of bool the carrier of (TOP-REAL 2)
(Ball (|[X,B]|,(uB * x9))) /\ () is Element of bool the carrier of (TOP-REAL 2)
r is Element of the carrier of ()
f is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
f `1 is V28() real ext-real Element of REAL
f `2 is V28() real ext-real Element of REAL
|[(f `1),(f `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
].V9,1.] is V169() V170() V171() V267() V272() Element of bool REAL
(X,B,uB) " ].V9,1.] is Element of bool the carrier of ()
(X,B,uB) . r is V28() real ext-real Element of the carrier of I[01]
|[X,B]| - f is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[X,B]|,f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[X,B]| - f).| is V28() real ext-real non negative Element of REAL
uB * V9 is V28() real ext-real set
|.(|[X,B]| - f).| - (uB * V9) is V28() real ext-real Element of REAL
f is non empty V28() real ext-real positive non negative set
Ball (f,f) is Element of bool the carrier of (TOP-REAL 2)
(Ball (f,f)) /\ () is Element of bool the carrier of (TOP-REAL 2)
u is Element of bool the carrier of ()
{ ((Ball (|[(f `1),(f `2)]|,b₁)) /\ ()) where b₁ is V28() real ext-real Element of REAL : not b₁ <= 0 } is set
V . |[(f `1),(f `2)]| is set
b is Element of bool the carrier of ()
Union V is open quasi_basis Element of bool (bool the carrier of ())
bool (bool the carrier of ()) is non empty set
|[X,B]| - f is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[X,B]|,f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[X,B]| - f).| is V28() real ext-real non negative Element of REAL
uB * V9 is V28() real ext-real set
|.(|[X,B]| - f).| - (uB * V9) is V28() real ext-real Element of REAL
(|.(|[X,B]| - f).| - (uB * V9)) / 2 is V28() real ext-real Element of REAL
{f} is functional non empty trivial finite V37() 1 -element Element of bool the carrier of (TOP-REAL 2)
f is non empty V28() real ext-real positive non negative set
|[(f `1),f]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[(f `1),f]|,f) is Element of bool the carrier of (TOP-REAL 2)
(Ball (|[(f `1),f]|,f)) \/ {f} is Element of bool the carrier of (TOP-REAL 2)
u is Element of bool the carrier of ()
|[(f `1),0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
{|[(f `1),0]|} is functional non empty trivial finite V37() 1 -element Element of bool the carrier of (TOP-REAL 2)
{ ((Ball (|[(f `1),b₁]|,b₁)) \/ {|[(f `1),0]|}) where b<sub>1</sub> is V28() real ext-real Element of REAL : not b<sub>1</sub> <= 0 } is set
V . |[(f `1),(f `2)]| is set
b is Element of bool the carrier of ()
Union V is open quasi_basis Element of bool (bool the carrier of ())
bool (bool the carrier of ()) is non empty set
Union V is open quasi_basis Element of bool (bool the carrier of ())
bool (bool the carrier of ()) is non empty set
Union V is open quasi_basis Element of bool (bool the carrier of ())
bool (bool the carrier of ()) is non empty set
X is Element of bool the carrier of ()
X ` is Element of bool the carrier of ()
the carrier of () \ X is set
B is V28() real ext-real Element of REAL
uB is V28() real ext-real Element of REAL
|[B,uB]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
x is non empty V28() real ext-real positive non negative set
Ball (|[B,uB]|,x) is Element of bool the carrier of (TOP-REAL 2)
(Ball (|[B,uB]|,x)) /\ () is Element of bool the carrier of (TOP-REAL 2)
V is non empty V28() real ext-real positive non negative set
(B,V,x) is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
V9 is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
V9 . |[B,uB]| is V28() real ext-real set
|[B,V]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[B,V]| `2 is V28() real ext-real Element of REAL
x9 is V28() real ext-real set
r is V28() real ext-real set
|[x9,r]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
V9 . |[x9,r]| is V28() real ext-real set
|[B,uB]| - |[x9,r]| is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(|[B,uB]|,|[x9,r]|) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(|[B,uB]| - |[x9,r]|).| is V28() real ext-real non negative Element of REAL
|.(|[B,uB]| - |[x9,r]|).| / x is V28() real ext-real non negative Element of REAL
B is Element of the carrier of ()
uB is Element of the carrier of ()
x is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
V is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
x - V is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(x,V) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(x - V).| is V28() real ext-real non negative Element of REAL
V9 is non empty V28() real ext-real positive non negative set
V9 / 2 is non empty V28() real ext-real positive non negative Element of REAL
r is open Element of bool the carrier of ()
x9 is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
f is open Element of bool the carrier of ()
f is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
V - x9 is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(V,x9) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(V - x9).| is V28() real ext-real non negative Element of REAL
(V9 / 2) + (V9 / 2) is non empty V28() real ext-real positive non negative Element of REAL
x - x9 is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(x,x9) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(x - x9).| is V28() real ext-real non negative Element of REAL
|.(x - x9).| + |.(V - x9).| is V28() real ext-real non negative Element of REAL
x9 - V is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(x9,V) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(x9 - V).| is V28() real ext-real non negative Element of REAL
|.(x - x9).| + |.(x9 - V).| is V28() real ext-real non negative Element of REAL
x - f is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(x,f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(x - f).| is V28() real ext-real non negative Element of REAL
f - V is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(f,V) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(f - V).| is V28() real ext-real non negative Element of REAL
|.(x - f).| + |.(f - V).| is V28() real ext-real non negative Element of REAL
V - f is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
K391(V,f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like V159() V160() V161() FinSequence of REAL
|.(V - f).| is V28() real ext-real non negative Element of REAL
|.(x - f).| + |.(V - f).| is V28() real ext-real non negative Element of REAL
B is Relation-like the carrier of () -defined Function-like total Neighborhood_System of ()
bool (bool the carrier of ()) is non empty set
Union B is open quasi_basis Element of bool (bool the carrier of ())
x is Element of the carrier of ()
V is Element of bool the carrier of ()
uB is open quasi_prebasis Element of bool (bool the carrier of ())
B . x is open V235((),x) Element of bool (bool the carrier of ())
V9 is Element of bool the carrier of ()
x9 is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
x9 `1 is V28() real ext-real Element of REAL
x9 `2 is V28() real ext-real Element of REAL
|[(x9 `1),(x9 `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
|[(x9 `1),0]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
{|[(x9 `1),0]|} is functional non empty trivial finite V37() 1 -element Element of bool the carrier of (TOP-REAL 2)
{ ((Ball (|[(x9 `1),b₁]|,b₁)) \/ {|[(x9 `1),0]|}) where b<sub>1</sub> is V28() real ext-real Element of REAL : not b<sub>1</sub> <= 0 } is set
r is V28() real ext-real Element of REAL
|[(x9 `1),r]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
Ball (|[(x9 `1),r]|,r) is Element of bool the carrier of (TOP-REAL 2)
(Ball (|[(x9 `1),r]|,r)) \/ {|[(x9 `1),0]|} is Element of bool the carrier of (TOP-REAL 2)
V9 ` is Element of bool the carrier of ()
the carrier of () \ V9 is set
V9 \ {|[(x9 `1),0]|} is Element of bool the carrier of ()
2 * r is V28() real ext-real Element of REAL
f is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
f . |[(x9 `1),0]| is V28() real ext-real set
f is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
f . x is V28() real ext-real Element of the carrier of I[01]
V ` is Element of bool the carrier of ()
the carrier of () \ V is set
f .: (V `) is V169() V170() V171() Element of bool the carrier of I[01]
u is set
b is Element of the carrier of ()
f . b is V28() real ext-real Element of the carrier of I[01]
c is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
c `1 is V28() real ext-real Element of REAL
c `2 is V28() real ext-real Element of REAL
|[(c `1),(c `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
{ ((Ball (|[(x9 `1),(x9 `2)]|,b₁)) /\ ()) where b₁ is V28() real ext-real Element of REAL : not b₁ <= 0 } is set
r is V28() real ext-real Element of REAL
Ball (|[(x9 `1),(x9 `2)]|,r) is Element of bool the carrier of (TOP-REAL 2)
(Ball (|[(x9 `1),(x9 `2)]|,r)) /\ () is Element of bool the carrier of (TOP-REAL 2)
V9 ` is Element of bool the carrier of ()
the carrier of () \ V9 is set
f is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
f . |[(x9 `1),(x9 `2)]| is V28() real ext-real set
f is Relation-like the carrier of () -defined the carrier of I[01] -valued Function-like non empty total quasi_total continuous V159() V160() V161() Element of bool [: the carrier of (), the carrier of I[01]:]
f . x is V28() real ext-real Element of the carrier of I[01]
V ` is Element of bool the carrier of ()
the carrier of () \ V is set
f .: (V `) is V169() V170() V171() Element of bool the carrier of I[01]
u is set
b is Element of the carrier of ()
f . b is V28() real ext-real Element of the carrier of I[01]
c is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)
c `1 is V28() real ext-real Element of REAL
c `2 is V28() real ext-real Element of REAL
|[(c `1),(c `2)]| is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like V159() V160() V161() Element of the carrier of (TOP-REAL 2)