:: SIMPLEX0 semantic presentation

REAL is set
NAT is non empty non trivial non empty-membered epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal Element of bool REAL
bool REAL is non empty subset-closed V120() V252() d.union-closed set
NAT is non empty non trivial non empty-membered epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal set
bool NAT is non empty non trivial non empty-membered non finite subset-closed V120() V252() d.union-closed set
K215() is TopStruct
the carrier of K215() is set
COMPLEX is set
RAT is set
INT is set
[:COMPLEX,COMPLEX:] is Relation-like set
bool [:COMPLEX,COMPLEX:] is non empty subset-closed V120() V252() d.union-closed set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is Relation-like set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty subset-closed V120() V252() d.union-closed set
[:REAL,REAL:] is Relation-like set
bool [:REAL,REAL:] is non empty subset-closed V120() V252() d.union-closed set
[:[:REAL,REAL:],REAL:] is Relation-like set
bool [:[:REAL,REAL:],REAL:] is non empty subset-closed V120() V252() d.union-closed set
[:RAT,RAT:] is Relation-like set
bool [:RAT,RAT:] is non empty subset-closed V120() V252() d.union-closed set
[:[:RAT,RAT:],RAT:] is Relation-like set
bool [:[:RAT,RAT:],RAT:] is non empty subset-closed V120() V252() d.union-closed set
[:INT,INT:] is Relation-like set
bool [:INT,INT:] is non empty subset-closed V120() V252() d.union-closed set
[:[:INT,INT:],INT:] is Relation-like set
bool [:[:INT,INT:],INT:] is non empty subset-closed V120() V252() d.union-closed set
[:NAT,NAT:] is non empty non trivial non empty-membered Relation-like non finite set
[:[:NAT,NAT:],NAT:] is non empty non trivial non empty-membered Relation-like non finite set
bool [:[:NAT,NAT:],NAT:] is non empty non trivial non empty-membered non finite subset-closed V120() V252() d.union-closed set
bool NAT is non empty non trivial non empty-membered non finite subset-closed V120() V252() d.union-closed set
K440() is set
{} is empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V17() real Relation-like non-empty empty-yielding Function-like one-to-one constant functional finite finite-yielding finite-membered cardinal {} -element integer Function-yielding V200() set
the empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V17() real Relation-like non-empty empty-yielding Function-like one-to-one constant functional finite finite-yielding finite-membered cardinal {} -element integer Function-yielding V200() set is empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V17() real Relation-like non-empty empty-yielding Function-like one-to-one constant functional finite finite-yielding finite-membered cardinal {} -element integer Function-yielding V200() set
{{}} is non empty trivial functional finite finite-membered 1 -element subset-closed V120() V252() d.union-closed set
1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V17() real finite cardinal integer Element of NAT
K387({{}}) is M22({{}})
[:K387({{}}),{{}}:] is Relation-like set
bool [:K387({{}}),{{}}:] is non empty subset-closed V120() V252() d.union-closed set
K89(K387({{}}),{{}}) is set
0 is empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V17() real Relation-like non-empty empty-yielding Function-like one-to-one constant functional finite finite-yielding finite-membered cardinal {} -element integer Function-yielding V200() Element of NAT
K63(1) is ext-real non positive V17() real integer Element of REAL
[:{{}},NAT:] is non empty non trivial non empty-membered Relation-like non finite set
NAT \/ [:{{}},NAT:] is non empty set
[{},{}] is set
{{},{}} is non empty functional finite finite-membered set
{{{},{}},{{}}} is non empty with_non-empty_elements non empty-membered finite finite-membered set
{[{},{}]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(NAT \/ [:{{}},NAT:]) \ {[{},{}]} is Element of bool (NAT \/ [:{{}},NAT:])
bool (NAT \/ [:{{}},NAT:]) is non empty subset-closed V120() V252() d.union-closed set
+infty is non empty ext-real positive non negative non real set
-infty is non empty ext-real non positive negative non real set
bool {} is non empty finite finite-membered subset-closed V120() V252() d.union-closed set
union {} is ext-real non negative epsilon-transitive epsilon-connected ordinal natural V17() real finite cardinal integer set
X is set
n is set
KX is set
X is set
X is set
n is set
X is set
Sub_of_Fin X is Element of bool X
bool X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
n is set
X is subset-closed V252() set
Sub_of_Fin X is finite-membered Element of bool X
bool X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
n is set
KX is set
X is set
n is set
bool n is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed Element of bool (bool n)
bool n is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool n) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
KX is set
n is set
KX is set
bool n is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed Element of bool (bool n)
bool n is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool n) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
X is subset-closed V252() set
n is subset-closed V252() set
X \/ n is set
KX is set
SX is set
X /\ n is set
KX is set
SX is set
X is set
{ (bool b₁) where b₁ is Element of X : b₁ in X } is set
union { (bool b₁) where b₁ is Element of X : b₁ in X } is set
KX is set
SX is set
P is set
n is Element of X
bool n is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed Element of bool (bool n)
bool n is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool n) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
KX is subset-closed V252() set
SX is set
bool SX is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed Element of bool (bool SX)
bool SX is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool SX) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
SX is set
P is set
n is set
Pn is Element of X
bool Pn is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed Element of bool (bool Pn)
bool Pn is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool Pn) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
X is set
{ (bool b₁) where b₁ is Element of X : b₁ in X } is set
union { (bool b₁) where b₁ is Element of X : b₁ in X } is set
n is subset-closed V252() set
n is subset-closed V252() set
KX is subset-closed V252() set
X is set
n is set
(n) is subset-closed V252() set
{ (bool b₁) where b₁ is Element of n : b₁ in n } is set
union { (bool b₁) where b₁ is Element of n : b₁ in n } is set
SX is subset-closed V252() set
P is set
n is Element of n
bool n is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed Element of bool (bool n)
bool n is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool n) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
Pn is set
Fm is set
P is set
bool P is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed Element of bool (bool P)
bool P is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool P) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
X is set
bool X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
n is Element of bool (bool X)
(n) is subset-closed V252() set
bool X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed Element of bool (bool X)
SX is set
P is set
({}) is subset-closed V252() set
X is set
n is set
X is non empty set
(X) is subset-closed V252() set
n is set
X is non empty non empty-membered set
(X) is non empty non with_non-empty_elements subset-closed V252() set
n is non empty set
X is finite-membered set
(X) is subset-closed V252() set
n is set
KX is set
X is set
(X) is subset-closed V252() set
n is set
KX is set
SX is set
X is set
{X} is non empty trivial finite 1 -element set
({X}) is non empty non with_non-empty_elements subset-closed V252() set
bool X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed Element of bool (bool X)
bool X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
KX is set
SX is set
KX is set
X is set
(X) is subset-closed V252() set
n is set
X \/ n is set
((X \/ n)) is subset-closed V252() set
(n) is subset-closed V252() set
(X) \/ (n) is subset-closed V252() set
n is set
Pn is set
n is set
Pn is set
Pn is set
X is set
(X) is subset-closed V252() set
n is set
(n) is subset-closed V252() set
P is set
n is set
Pn is set
P is set
n is set
X is set
(X) is subset-closed V252() set
KX is set
SX is set
X is set
(X) is subset-closed V252() set
KX is set
SX is set
X is set
bool X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
n is set
KX is set
card KX is epsilon-transitive epsilon-connected ordinal cardinal set
{ b₁ where b₁ is Element of bool X : ( b₁ in n & card b₁ c= card KX ) } is set
bool X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed Element of bool (bool X)
P is set
n is Element of bool X
card n is epsilon-transitive epsilon-connected ordinal cardinal set
P is Element of bool (bool X)
n is Element of bool X
card n is epsilon-transitive epsilon-connected ordinal cardinal set
Pn is Element of bool X
card Pn is epsilon-transitive epsilon-connected ordinal cardinal set
SX is Element of bool (bool X)
P is Element of bool (bool X)
n is set
card n is epsilon-transitive epsilon-connected ordinal cardinal set
n is set
card n is epsilon-transitive epsilon-connected ordinal cardinal set
X is non empty set
bool X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
n is set
{n} is non empty trivial finite 1 -element set
bool {n} is non empty non with_non-empty_elements finite finite-membered subset-closed V120() V252() d.union-closed Element of bool (bool {n})
bool {n} is non empty non with_non-empty_elements finite finite-membered subset-closed V120() V252() d.union-closed set
bool (bool {n}) is non empty non with_non-empty_elements finite finite-membered subset-closed V120() V252() d.union-closed set
KX is Element of bool (bool X)
X is set
n is set
KX is finite set
(X,n,KX) is Element of bool (bool X)
bool X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
SX is set
card SX is epsilon-transitive epsilon-connected ordinal cardinal set
card KX is ext-real non negative epsilon-transitive epsilon-connected ordinal natural V17() real finite cardinal integer Element of NAT
X is set
n is subset-closed V252() set
KX is set
(X,n,KX) is Element of bool (bool X)
bool X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
SX is set
P is set
card SX is epsilon-transitive epsilon-connected ordinal cardinal set
card KX is epsilon-transitive epsilon-connected ordinal cardinal set
card P is epsilon-transitive epsilon-connected ordinal cardinal set
X is set
n is non empty non with_non-empty_elements set
KX is set
(X,n,KX) is Element of bool (bool X)
bool X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
card {} is empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V17() real Relation-like non-empty empty-yielding Function-like one-to-one constant functional finite finite-yielding finite-membered cardinal {} -element integer subset-closed Function-yielding V200() V252() Element of NAT
card KX is epsilon-transitive epsilon-connected ordinal cardinal set
X is non empty set
bool X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
n is non empty non with_non-empty_elements non empty-membered subset-closed V252() Element of bool (bool X)
KX is non empty set
(X,n,KX) is non empty non with_non-empty_elements subset-closed V252() Element of bool (bool X)
SX is non empty set
P is set
card KX is non empty epsilon-transitive epsilon-connected ordinal cardinal set
{P} is non empty trivial finite 1 -element set
card {P} is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V17() real finite cardinal integer Element of NAT
X is set
bool X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
X is set
union X is set
n is set
KX is set
SX is set
X is c=-linear finite set
card X is ext-real non negative epsilon-transitive epsilon-connected ordinal natural V17() real finite cardinal integer Element of NAT
union X is set
card (union X) is epsilon-transitive epsilon-connected ordinal cardinal set
n is ext-real non negative epsilon-transitive epsilon-connected ordinal natural V17() real finite cardinal integer set
n + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V17() real finite cardinal integer Element of NAT
KX is ext-real non negative epsilon-transitive epsilon-connected ordinal natural V17() real finite cardinal integer set
SX is c=-linear finite set
union SX is set
card (union SX) is epsilon-transitive epsilon-connected ordinal cardinal set
card SX is ext-real non negative epsilon-transitive epsilon-connected ordinal natural V17() real finite cardinal integer Element of NAT
P is finite set
{P} is non empty trivial finite finite-membered 1 -element set
SX \ {P} is c=-linear finite Element of bool SX
bool SX is non empty non with_non-empty_elements finite finite-membered subset-closed V120() V252() d.union-closed set
n is c=-linear finite set
n \/ {P} is non empty finite set
union n is set
union {P} is finite set
(union n) \/ (union {P}) is set
(union n) \/ P is set
Pn is finite set
card Pn is ext-real non negative epsilon-transitive epsilon-connected ordinal natural V17() real finite cardinal integer Element of NAT
card n is ext-real non negative epsilon-transitive epsilon-connected ordinal natural V17() real finite cardinal integer Element of NAT
card P is ext-real non negative epsilon-transitive epsilon-connected ordinal natural V17() real finite cardinal integer Element of NAT
1 + (card n) is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V17() real finite cardinal integer Element of NAT
n is ext-real non negative epsilon-transitive epsilon-connected ordinal natural V17() real finite cardinal integer set
KX is c=-linear finite set
union KX is set
card (union KX) is epsilon-transitive epsilon-connected ordinal cardinal set
card KX is ext-real non negative epsilon-transitive epsilon-connected ordinal natural V17() real finite cardinal integer Element of NAT
X is set
union X is set
n is set
(union X) \/ n is set
{((union X) \/ n)} is non empty trivial finite 1 -element set
X \/ {((union X) \/ n)} is non empty set
P is set
n is set
X is non empty set
bool X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
card X is non empty epsilon-transitive epsilon-connected ordinal cardinal set
n is Relation-like set
n |_2 X is Relation-like set
bool X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed Element of bool (bool X)
SX is set
(n |_2 X) -Seg SX is set
X \ ((n |_2 X) -Seg SX) is Element of bool X
[:X,(bool X):] is non empty Relation-like set
bool [:X,(bool X):] is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
SX is Relation-like X -defined bool X -valued Function-like V33(X, bool X) Element of bool [:X,(bool X):]
bool (bool X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
rng SX is Element of bool (bool X)
P is Element of bool (bool X)
card P is epsilon-transitive epsilon-connected ordinal cardinal set
dom SX is Element of bool X
n is set
SX . n is set
(n |_2 X) -Seg n is set
X \ ((n |_2 X) -Seg n) is Element of bool X
n is set
Pn is set
Fm is set
SX . Fm is set
m is set
SX . m is set
(n |_2 X) -Seg Fm is set
(n |_2 X) -Seg m is set
X \ ((n |_2 X) -Seg m) is Element of bool X
X \ ((n |_2 X) -Seg Fm) is Element of bool X
field (n |_2 X) is set
n is set
(n |_2 X) -Seg n is set
Pn is set
[Pn,n] is set
{Pn,n} is non empty finite set
{Pn} is non empty trivial finite 1 -element set
{{Pn,n},{Pn}} is non empty with_non-empty_elements non empty-membered finite finite-membered set
[n,Pn] is set
{n,Pn} is non empty finite set
{n} is non empty trivial finite 1 -element set
{{n,Pn},{n}} is non empty with_non-empty_elements non empty-membered finite finite-membered set
SX . n is set
X \ ((n |_2 X) -Seg n) is Element of bool X
Pn is set
(n |_2 X) -Seg Pn is set
Fm is set
(n |_2 X) -Seg Fm is set
SX . Pn is set
SX . Fm is set
m is Element of bool X
m ` is Element of bool X
X \ m is set
SX is Element of bool X
SX ` is Element of bool X
X \ SX is set
[Pn,Fm] is set
{Pn,Fm} is non empty finite set
{Pn} is non empty trivial finite 1 -element set
{{Pn,Fm},{Pn}} is non empty with_non-empty_elements non empty-membered finite finite-membered set
[Fm,Pn] is set
{Fm,Pn} is non empty finite set
{Fm} is non empty trivial finite 1 -element set
{{Fm,Pn},{Fm}} is non empty with_non-empty_elements non empty-membered finite finite-membered set
Pn is set
card Pn is epsilon-transitive epsilon-connected ordinal cardinal set
Fm is set
SX . Fm is set
{Fm} is non empty trivial finite 1 -element set
Pn \ {Fm} is Element of bool Pn
bool Pn is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
(n |_2 X) -Seg Fm is set
X \ ((n |_2 X) -Seg Fm) is Element of bool X
((n |_2 X) -Seg Fm) \/ {Fm} is non empty set
m is Element of bool X
X \ m is Element of bool X
m \ ((n |_2 X) -Seg Fm) is Element of bool X
{Fm} \ ((n |_2 X) -Seg Fm) is trivial finite Element of bool {Fm}
bool {Fm} is non empty non with_non-empty_elements finite finite-membered subset-closed V120() V252() d.union-closed set
m ` is Element of bool X
X \ m is set
SX is set
(n |_2 X) -Seg SX is set
[Fm,SX] is set
{Fm,SX} is non empty finite set
{{Fm,SX},{Fm}} is non empty with_non-empty_elements non empty-membered finite finite-membered set
[SX,Fm] is set
{SX,Fm} is non empty finite set
{SX} is non empty trivial finite 1 -element set
{{SX,Fm},{SX}} is non empty with_non-empty_elements non empty-membered finite finite-membered set
SX . SX is set
(X \ ((n |_2 X) -Seg Fm)) \ {Fm} is Element of bool X
X is set
bool X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
card X is epsilon-transitive epsilon-connected ordinal cardinal set
n is Element of bool (bool X)
KX is set
{ b₁ where b₁ is Element of bool X : ( KX in b₁ & b₁ in n ) } is set
P is set
n is Element of bool X
P is Element of bool (bool X)
n is set
Pn is Element of bool X
Pn is set
KX is Relation-like Function-like set
dom KX is set
bool X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed Element of bool (bool X)
SX is set
{ b₁ where b₁ is Element of bool X : ( b₁ c< SX & b₁ in n ) } is set
n is set
Pn is Element of bool X
n is Element of bool (bool X)
union n is Element of bool X
Pn is Element of bool X
Fm is set
Fm is set
Fm is set
Fm is set
m is Element of bool X
Fm is set
[:n,(bool X):] is Relation-like set
bool [:n,(bool X):] is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
SX is Relation-like n -defined bool X -valued Function-like V33(n, bool X) Element of bool [:n,(bool X):]
P is set
SX . P is set
P \ (SX . P) is Element of bool P
bool P is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (P \ (SX . P)) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed Element of bool (bool (P \ (SX . P)))
bool (P \ (SX . P)) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool (P \ (SX . P))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
[:(P \ (SX . P)),(bool (P \ (SX . P))):] is Relation-like set
bool [:(P \ (SX . P)),(bool (P \ (SX . P))):] is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
card (P \ (SX . P)) is epsilon-transitive epsilon-connected ordinal cardinal set
n is Element of bool (bool (P \ (SX . P)))
card n is epsilon-transitive epsilon-connected ordinal cardinal set
Pn is Relation-like Function-like set
dom Pn is set
rng Pn is set
Fm is Relation-like P \ (SX . P) -defined bool (P \ (SX . P)) -valued Function-like V33(P \ (SX . P), bool (P \ (SX . P))) Element of bool [:(P \ (SX . P)),(bool (P \ (SX . P))):]
P is Relation-like Function-like set
dom P is set
n is set
SX . n is set
n \ (SX . n) is Element of bool n
bool n is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (n \ (SX . n)) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed Element of bool (bool (n \ (SX . n)))
bool (n \ (SX . n)) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool (n \ (SX . n))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
[:(n \ (SX . n)),(bool (n \ (SX . n))):] is Relation-like set
bool [:(n \ (SX . n)),(bool (n \ (SX . n))):] is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
P . n is set
Pn is Relation-like n \ (SX . n) -defined bool (n \ (SX . n)) -valued Function-like V33(n \ (SX . n), bool (n \ (SX . n))) Element of bool [:(n \ (SX . n)),(bool (n \ (SX . n))):]
rng Pn is Element of bool (bool (n \ (SX . n)))
bool (bool (n \ (SX . n))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
dom Pn is Element of bool (n \ (SX . n))
n is Relation-like Function-like Function-yielding V200() set
Pn is set
KX . Pn is set
SX . (KX . Pn) is set
(KX . Pn) \ (SX . (KX . Pn)) is Element of bool (KX . Pn)
bool (KX . Pn) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
Pn is set
KX . Pn is set
SX . (KX . Pn) is set
n . (KX . Pn) is Relation-like Function-like set
(n . (KX . Pn)) . Pn is set
(SX . (KX . Pn)) \/ ((n . (KX . Pn)) . Pn) is set
(KX . Pn) \ (SX . (KX . Pn)) is Element of bool (KX . Pn)
bool (KX . Pn) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool ((KX . Pn) \ (SX . (KX . Pn))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed Element of bool (bool ((KX . Pn) \ (SX . (KX . Pn))))
bool ((KX . Pn) \ (SX . (KX . Pn))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool ((KX . Pn) \ (SX . (KX . Pn)))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
[:((KX . Pn) \ (SX . (KX . Pn))),(bool ((KX . Pn) \ (SX . (KX . Pn)))):] is Relation-like set
bool [:((KX . Pn) \ (SX . (KX . Pn))),(bool ((KX . Pn) \ (SX . (KX . Pn)))):] is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
m is Relation-like (KX . Pn) \ (SX . (KX . Pn)) -defined bool ((KX . Pn) \ (SX . (KX . Pn))) -valued Function-like V33((KX . Pn) \ (SX . (KX . Pn)), bool ((KX . Pn) \ (SX . (KX . Pn)))) Element of bool [:((KX . Pn) \ (SX . (KX . Pn))),(bool ((KX . Pn) \ (SX . (KX . Pn)))):]
rng m is Element of bool (bool ((KX . Pn) \ (SX . (KX . Pn))))
bool (bool ((KX . Pn) \ (SX . (KX . Pn)))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
dom m is Element of bool ((KX . Pn) \ (SX . (KX . Pn)))
m . Pn is set
[:X,(bool X):] is Relation-like set
bool [:X,(bool X):] is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
Pn is Relation-like X -defined bool X -valued Function-like V33(X, bool X) Element of bool [:X,(bool X):]
dom Pn is Element of bool X
rng Pn is Element of bool (bool X)
bool (bool X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
Fm is set
SX . Fm is set
Fm \ (SX . Fm) is Element of bool Fm
bool Fm is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (Fm \ (SX . Fm)) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed Element of bool (bool (Fm \ (SX . Fm)))
bool (Fm \ (SX . Fm)) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool (Fm \ (SX . Fm))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
[:(Fm \ (SX . Fm)),(bool (Fm \ (SX . Fm))):] is Relation-like set
bool [:(Fm \ (SX . Fm)),(bool (Fm \ (SX . Fm))):] is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
n . Fm is Relation-like Function-like set
m is Relation-like Fm \ (SX . Fm) -defined bool (Fm \ (SX . Fm)) -valued Function-like V33(Fm \ (SX . Fm), bool (Fm \ (SX . Fm))) Element of bool [:(Fm \ (SX . Fm)),(bool (Fm \ (SX . Fm))):]
rng m is Element of bool (bool (Fm \ (SX . Fm)))
bool (bool (Fm \ (SX . Fm))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
dom m is Element of bool (Fm \ (SX . Fm))
SX is set
m . SX is set
KX . SX is set
Pn . SX is set
(SX . Fm) \/ (Fm \ (SX . Fm)) is set
Fm is set
card Fm is epsilon-transitive epsilon-connected ordinal cardinal set
m is set
Pn . m is set
KX . m is set
SX . (KX . m) is set
(KX . m) \ (SX . (KX . m)) is Element of bool (KX . m)
bool (KX . m) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool ((KX . m) \ (SX . (KX . m))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed Element of bool (bool ((KX . m) \ (SX . (KX . m))))
bool ((KX . m) \ (SX . (KX . m))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool ((KX . m) \ (SX . (KX . m)))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
[:((KX . m) \ (SX . (KX . m))),(bool ((KX . m) \ (SX . (KX . m)))):] is Relation-like set
bool [:((KX . m) \ (SX . (KX . m))),(bool ((KX . m) \ (SX . (KX . m)))):] is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
n . (KX . m) is Relation-like Function-like set
c₁₁ is Relation-like (KX . m) \ (SX . (KX . m)) -defined bool ((KX . m) \ (SX . (KX . m))) -valued Function-like V33((KX . m) \ (SX . (KX . m)), bool ((KX . m) \ (SX . (KX . m)))) Element of bool [:((KX . m) \ (SX . (KX . m))),(bool ((KX . m) \ (SX . (KX . m)))):]
rng c₁₁ is Element of bool (bool ((KX . m) \ (SX . (KX . m))))
bool (bool ((KX . m) \ (SX . (KX . m)))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
dom c₁₁ is Element of bool ((KX . m) \ (SX . (KX . m)))
c₁₁ . m is set
(SX . (KX . m)) \/ (c₁₁ . m) is set
card (c₁₁ . m) is epsilon-transitive epsilon-connected ordinal cardinal set
TOP1 is set
{TOP1} is non empty trivial finite 1 -element set
Fm \ {TOP1} is Element of bool Fm
bool Fm is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
((SX . (KX . m)) \/ (c₁₁ . m)) \ (c₁₁ . m) is Element of bool ((SX . (KX . m)) \/ (c₁₁ . m))
bool ((SX . (KX . m)) \/ (c₁₁ . m)) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
(SX . (KX . m)) \ (c₁₁ . m) is Element of bool (SX . (KX . m))
bool (SX . (KX . m)) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
card (c₁₁ . m) is epsilon-transitive epsilon-connected ordinal cardinal set
TOP1 is set
{TOP1} is non empty trivial finite 1 -element set
(c₁₁ . m) \ {TOP1} is Element of bool (c₁₁ . m)
bool (c₁₁ . m) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
Fm \ {TOP1} is Element of bool Fm
bool Fm is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
K2 is set
c₁₁ . K2 is set
KX . K2 is set
Pn . K2 is set
(SX . (KX . m)) \/ ((c₁₁ . m) \ {TOP1}) is set
((SX . (KX . m)) \/ (c₁₁ . m)) \ {TOP1} is Element of bool ((SX . (KX . m)) \/ (c₁₁ . m))
bool ((SX . (KX . m)) \/ (c₁₁ . m)) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
card (c₁₁ . m) is epsilon-transitive epsilon-connected ordinal cardinal set
Fm is set
m is set
SX is set
Pn . SX is set
c₁₁ is set
Pn . c₁₁ is set
KX . SX is set
KX . c₁₁ is set
SX . (KX . SX) is set
(KX . SX) \ (SX . (KX . SX)) is Element of bool (KX . SX)
bool (KX . SX) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool ((KX . SX) \ (SX . (KX . SX))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed Element of bool (bool ((KX . SX) \ (SX . (KX . SX))))
bool ((KX . SX) \ (SX . (KX . SX))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool ((KX . SX) \ (SX . (KX . SX)))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
[:((KX . SX) \ (SX . (KX . SX))),(bool ((KX . SX) \ (SX . (KX . SX)))):] is Relation-like set
bool [:((KX . SX) \ (SX . (KX . SX))),(bool ((KX . SX) \ (SX . (KX . SX)))):] is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
n . (KX . SX) is Relation-like Function-like set
TOP2 is Relation-like (KX . SX) \ (SX . (KX . SX)) -defined bool ((KX . SX) \ (SX . (KX . SX))) -valued Function-like V33((KX . SX) \ (SX . (KX . SX)), bool ((KX . SX) \ (SX . (KX . SX)))) Element of bool [:((KX . SX) \ (SX . (KX . SX))),(bool ((KX . SX) \ (SX . (KX . SX)))):]
rng TOP2 is Element of bool (bool ((KX . SX) \ (SX . (KX . SX))))
bool (bool ((KX . SX) \ (SX . (KX . SX)))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
dom TOP2 is Element of bool ((KX . SX) \ (SX . (KX . SX)))
TOP2 . SX is set
(SX . (KX . SX)) \/ (TOP2 . SX) is set
SX . (KX . c₁₁) is set
(KX . c₁₁) \ (SX . (KX . c₁₁)) is Element of bool (KX . c₁₁)
bool (KX . c₁₁) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool ((KX . c₁₁) \ (SX . (KX . c₁₁))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed Element of bool (bool ((KX . c₁₁) \ (SX . (KX . c₁₁))))
bool ((KX . c₁₁) \ (SX . (KX . c₁₁))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool ((KX . c₁₁) \ (SX . (KX . c₁₁)))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
[:((KX . c₁₁) \ (SX . (KX . c₁₁))),(bool ((KX . c₁₁) \ (SX . (KX . c₁₁)))):] is Relation-like set
bool [:((KX . c₁₁) \ (SX . (KX . c₁₁))),(bool ((KX . c₁₁) \ (SX . (KX . c₁₁)))):] is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
n . (KX . c₁₁) is Relation-like Function-like set
H2 is Relation-like (KX . c₁₁) \ (SX . (KX . c₁₁)) -defined bool ((KX . c₁₁) \ (SX . (KX . c₁₁))) -valued Function-like V33((KX . c₁₁) \ (SX . (KX . c₁₁)), bool ((KX . c₁₁) \ (SX . (KX . c₁₁)))) Element of bool [:((KX . c₁₁) \ (SX . (KX . c₁₁))),(bool ((KX . c₁₁) \ (SX . (KX . c₁₁)))):]
rng H2 is Element of bool (bool ((KX . c₁₁) \ (SX . (KX . c₁₁))))
bool (bool ((KX . c₁₁) \ (SX . (KX . c₁₁)))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
dom H2 is Element of bool ((KX . c₁₁) \ (SX . (KX . c₁₁)))
H2 . c₁₁ is set
(SX . (KX . c₁₁)) \/ (H2 . c₁₁) is set
TOP2 . c₁₁ is set
(SX . (KX . SX)) \/ ((KX . SX) \ (SX . (KX . SX))) is set
(SX . (KX . c₁₁)) \/ ((KX . c₁₁) \ (SX . (KX . c₁₁))) is set
Fm is set
Pn . Fm is set
KX . Fm is set
SX . (KX . Fm) is set
(KX . Fm) \ (SX . (KX . Fm)) is Element of bool (KX . Fm)
bool (KX . Fm) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool ((KX . Fm) \ (SX . (KX . Fm))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed Element of bool (bool ((KX . Fm) \ (SX . (KX . Fm))))
bool ((KX . Fm) \ (SX . (KX . Fm))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool ((KX . Fm) \ (SX . (KX . Fm)))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
[:((KX . Fm) \ (SX . (KX . Fm))),(bool ((KX . Fm) \ (SX . (KX . Fm)))):] is Relation-like set
bool [:((KX . Fm) \ (SX . (KX . Fm))),(bool ((KX . Fm) \ (SX . (KX . Fm)))):] is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
n . (KX . Fm) is Relation-like Function-like set
SX is Relation-like (KX . Fm) \ (SX . (KX . Fm)) -defined bool ((KX . Fm) \ (SX . (KX . Fm))) -valued Function-like V33((KX . Fm) \ (SX . (KX . Fm)), bool ((KX . Fm) \ (SX . (KX . Fm)))) Element of bool [:((KX . Fm) \ (SX . (KX . Fm))),(bool ((KX . Fm) \ (SX . (KX . Fm)))):]
rng SX is Element of bool (bool ((KX . Fm) \ (SX . (KX . Fm))))
bool (bool ((KX . Fm) \ (SX . (KX . Fm)))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
dom SX is Element of bool ((KX . Fm) \ (SX . (KX . Fm)))
SX . Fm is set
(SX . (KX . Fm)) \/ (SX . Fm) is set
card (rng Pn) is epsilon-transitive epsilon-connected ordinal cardinal set
Fm is set
m is set
Pn . Fm is set
Pn . m is set
KX . Fm is set
KX . m is set
SX . (KX . Fm) is set
n . (KX . Fm) is Relation-like Function-like set
(n . (KX . Fm)) . Fm is set
(SX . (KX . Fm)) \/ ((n . (KX . Fm)) . Fm) is set
SX . (KX . m) is set
n . (KX . m) is Relation-like Function-like set
(n . (KX . m)) . m is set
(SX . (KX . m)) \/ ((n . (KX . m)) . m) is set
(KX . m) \ (SX . (KX . m)) is Element of bool (KX . m)
bool (KX . m) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool ((KX . m) \ (SX . (KX . m))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed Element of bool (bool ((KX . m) \ (SX . (KX . m))))
bool ((KX . m) \ (SX . (KX . m))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool ((KX . m) \ (SX . (KX . m)))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
[:((KX . m) \ (SX . (KX . m))),(bool ((KX . m) \ (SX . (KX . m)))):] is Relation-like set
bool [:((KX . m) \ (SX . (KX . m))),(bool ((KX . m) \ (SX . (KX . m)))):] is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
TOP1 is Relation-like (KX . m) \ (SX . (KX . m)) -defined bool ((KX . m) \ (SX . (KX . m))) -valued Function-like V33((KX . m) \ (SX . (KX . m)), bool ((KX . m) \ (SX . (KX . m)))) Element of bool [:((KX . m) \ (SX . (KX . m))),(bool ((KX . m) \ (SX . (KX . m)))):]
rng TOP1 is Element of bool (bool ((KX . m) \ (SX . (KX . m))))
bool (bool ((KX . m) \ (SX . (KX . m)))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
dom TOP1 is Element of bool ((KX . m) \ (SX . (KX . m)))
TOP1 . m is set
(KX . Fm) \ (SX . (KX . Fm)) is Element of bool (KX . Fm)
bool (KX . Fm) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool ((KX . Fm) \ (SX . (KX . Fm))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed Element of bool (bool ((KX . Fm) \ (SX . (KX . Fm))))
bool ((KX . Fm) \ (SX . (KX . Fm))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool ((KX . Fm) \ (SX . (KX . Fm)))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
[:((KX . Fm) \ (SX . (KX . Fm))),(bool ((KX . Fm) \ (SX . (KX . Fm)))):] is Relation-like set
bool [:((KX . Fm) \ (SX . (KX . Fm))),(bool ((KX . Fm) \ (SX . (KX . Fm)))):] is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
K2 is Relation-like (KX . Fm) \ (SX . (KX . Fm)) -defined bool ((KX . Fm) \ (SX . (KX . Fm))) -valued Function-like V33((KX . Fm) \ (SX . (KX . Fm)), bool ((KX . Fm) \ (SX . (KX . Fm)))) Element of bool [:((KX . Fm) \ (SX . (KX . Fm))),(bool ((KX . Fm) \ (SX . (KX . Fm)))):]
rng K2 is Element of bool (bool ((KX . Fm) \ (SX . (KX . Fm))))
bool (bool ((KX . Fm) \ (SX . (KX . Fm)))) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
dom K2 is Element of bool ((KX . Fm) \ (SX . (KX . Fm)))
K2 . Fm is set
K2 . m is set
(SX . (KX . Fm)) \/ ((KX . Fm) \ (SX . (KX . Fm))) is set
(SX . (KX . m)) \/ ((KX . m) \ (SX . (KX . m))) is set
Fm is set
card Fm is epsilon-transitive epsilon-connected ordinal cardinal set
X is TopStruct
the carrier of X is set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
X is TopStruct
the carrier of X is set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool the carrier of X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
X is TopStruct
the carrier of X is set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool the carrier of X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed Element of bool (bool the carrier of X)
bool (bool the carrier of X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
{} (bool the carrier of X) is empty proper ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V17() real Relation-like non-empty empty-yielding Function-like one-to-one constant functional finite finite-yielding finite-membered cardinal {} -element integer subset-closed Function-yielding V200() V252() Element of bool (bool the carrier of X)
n is empty proper ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V17() real Relation-like non-empty empty-yielding Function-like one-to-one constant functional finite finite-yielding finite-membered cardinal {} -element integer subset-closed Function-yielding V200() V252() Element of bool (bool the carrier of X)
KX is Element of bool the carrier of X
X is TopStruct
the carrier of X is set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool the carrier of X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
the topology of X is Element of bool (bool the carrier of X)
n is Element of bool (bool the carrier of X)
KX is set
SX is Element of bool the carrier of X
KX is Element of bool the carrier of X
X is TopStruct
the carrier of X is set
X is TopStruct
the carrier of X is set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
{ b₁ where b₁ is Element of the carrier of X : b₁ is (X) } is set
[#] X is non proper Element of bool the carrier of X
KX is set
SX is Element of the carrier of X
P is Element of bool the carrier of X
KX is Element of bool the carrier of X
SX is Element of the carrier of X
P is Element of the carrier of X
n is Element of bool the carrier of X
KX is Element of bool the carrier of X
SX is set
P is Element of the carrier of X
SX is set
P is Element of the carrier of X
X is set
the empty trivial finite {} -element T_0 finite-membered finite-degree TopStruct is empty trivial finite {} -element T_0 finite-membered finite-degree TopStruct
[#] the empty trivial finite {} -element T_0 finite-membered finite-degree TopStruct is empty trivial non proper ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V17() real Relation-like non-empty empty-yielding Function-like one-to-one constant functional finite finite-yielding finite-membered cardinal {} -element integer subset-closed Function-yielding V200() V252() Element of bool the carrier of the empty trivial finite {} -element T_0 finite-membered finite-degree TopStruct
the carrier of the empty trivial finite {} -element T_0 finite-membered finite-degree TopStruct is empty trivial ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V17() real Relation-like non-empty empty-yielding Function-like one-to-one constant functional finite finite-yielding finite-membered cardinal {} -element integer subset-closed Function-yielding V200() V252() set
bool the carrier of the empty trivial finite {} -element T_0 finite-membered finite-degree TopStruct is non empty non with_non-empty_elements finite finite-membered subset-closed V120() V252() d.union-closed set
X is set
X is TopStruct
(X) is Element of bool the carrier of X
the carrier of X is set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
the topology of X is Element of bool (bool the carrier of X)
bool (bool the carrier of X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
n is non empty set
the Element of n is Element of n
KX is Element of bool the carrier of X
SX is Element of the carrier of X
n is set
KX is Element of the carrier of X
SX is Element of bool the carrier of X
X is set
n is TopStruct
the carrier of n is set
bool the carrier of n is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
(n) is Element of bool the carrier of n
KX is Element of bool the carrier of n
SX is Element of the carrier of n
X is TopStruct
the carrier of X is set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
(X) is Element of bool the carrier of X
n is Element of bool the carrier of X
KX is set
X is TopStruct
(X) is Element of bool the carrier of X
the carrier of X is set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
the topology of X is Element of bool (bool the carrier of X)
bool (bool the carrier of X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
union the topology of X is Element of bool the carrier of X
n is set
KX is Element of the carrier of X
SX is Element of bool the carrier of X
n is set
KX is set
SX is Element of bool the carrier of X
X is TopStruct
the topology of X is Element of bool (bool the carrier of X)
the carrier of X is set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool the carrier of X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
(X) is Element of bool the carrier of X
union the topology of X is Element of bool the carrier of X
bool (X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed Element of bool (bool (X))
bool (X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool (X)) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
X is TopStruct
the topology of X is Element of bool (bool the carrier of X)
the carrier of X is set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool the carrier of X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
X is TopStruct
the topology of X is Element of bool (bool the carrier of X)
the carrier of X is set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool the carrier of X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
X is TopStruct
the topology of X is Element of bool (bool the carrier of X)
the carrier of X is set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool the carrier of X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
X is TopStruct
the_family_of X is Element of bool (bool the carrier of X)
the carrier of X is set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool the carrier of X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
the topology of X is Element of bool (bool the carrier of X)
n is set
X is TopStruct
the topology of X is Element of bool (bool the carrier of X)
the carrier of X is set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool the carrier of X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
n is set
the_family_of X is Element of bool (bool the carrier of X)
KX is set
(X) is Element of bool the carrier of X
X is TopStruct
the carrier of X is set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
(X) is Element of bool the carrier of X
KX is Element of (X)
{ b₁ where b₁ is Element of bool the carrier of X : ( b₁ is open & KX in b₁ ) } is set
the topology of X is Element of bool (bool the carrier of X)
bool (bool the carrier of X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
SX is set
P is Element of bool the carrier of X
KX is set
the_family_of X is Element of bool (bool the carrier of X)
bool (bool the carrier of X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
the topology of X is Element of bool (bool the carrier of X)
SX is Element of bool the carrier of X
n is finite Element of bool the carrier of X
n is finite Element of bool the carrier of X
card n is ext-real non negative epsilon-transitive epsilon-connected ordinal natural V17() real finite cardinal integer Element of NAT
KX is finite Element of bool the carrier of X
card KX is ext-real non negative epsilon-transitive epsilon-connected ordinal natural V17() real finite cardinal integer Element of NAT
X is TopStruct
the_family_of X is Element of bool (bool the carrier of X)
the carrier of X is set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool the carrier of X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
the topology of X is Element of bool (bool the carrier of X)
n is set
KX is Element of bool the carrier of X
SX is set
P is Element of the carrier of X
(X) is Element of bool the carrier of X
n is Element of (X)
{n} is non empty trivial finite 1 -element set
KX \ {n} is Element of bool the carrier of X
{ b₁ where b₁ is Element of bool the carrier of X : ( n in b₁ & b₁ c= KX ) } is set
bool (KX \ {n}) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (KX \ {n}) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed Element of bool (bool (KX \ {n}))
bool (bool (KX \ {n})) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
{ b₁ where b₁ is Element of bool (KX \ {n}) : ex b₂ being Element of bool the carrier of X st
( S₁[b₂,b₁] & b₂ in { b₁ where b₃ is Element of bool the carrier of X : ( n in b₁ & b₁ c= KX ) } ) } is set
SX is set
c₁₁ is Element of bool the carrier of X
c₁₁ \/ {n} is non empty set
(c₁₁ \/ {n}) \ {n} is Element of bool (c₁₁ \/ {n})
bool (c₁₁ \/ {n}) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
{ b₁ where b₁ is Element of bool the carrier of X : ( b₁ is open & n in b₁ ) } is set
c₁₁ is set
TOP1 is Element of bool the carrier of X
c₁₁ is Element of bool the carrier of X
c₁₁ \ {n} is Element of bool c₁₁
bool c₁₁ is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
TOP1 is Element of bool (KX \ {n})
K2 is Element of bool (KX \ {n})
(KX \ {n}) \/ {n} is non empty set
X is set
bool {} is non empty non with_non-empty_elements finite finite-membered subset-closed V120() V252() d.union-closed Element of bool (bool {})
bool (bool {}) is non empty non with_non-empty_elements finite finite-membered subset-closed V120() V252() d.union-closed set
{} (bool {}) is empty proper ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V17() real Relation-like non-empty empty-yielding Function-like one-to-one constant functional finite finite-yielding finite-membered cardinal {} -element integer subset-closed Function-yielding V200() V252() Element of bool (bool {})
bool (bool {}) is non empty non with_non-empty_elements finite finite-membered subset-closed V120() V252() d.union-closed set
TopStruct(# {},({} (bool {})) #) is strict TopStruct
[#] TopStruct(# {},({} (bool {})) #) is non proper Element of bool the carrier of TopStruct(# {},({} (bool {})) #)
the carrier of TopStruct(# {},({} (bool {})) #) is set
bool the carrier of TopStruct(# {},({} (bool {})) #) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
n is (X)
X is non empty set
bool X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool {} is non empty non with_non-empty_elements finite finite-membered subset-closed V120() V252() d.union-closed Element of bool (bool {})
bool (bool {}) is non empty non with_non-empty_elements finite finite-membered subset-closed V120() V252() d.union-closed set
n is Element of bool (bool X)
TopStruct(# X,n #) is non empty strict TopStruct
[#] TopStruct(# X,n #) is non empty non proper Element of bool the carrier of TopStruct(# X,n #)
the carrier of TopStruct(# X,n #) is non empty set
bool the carrier of TopStruct(# X,n #) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
KX is (X)
SX is non empty set
[#] KX is non proper Element of bool the carrier of KX
the carrier of KX is set
bool the carrier of KX is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
the Element of X is Element of X
bool X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
{ the Element of X} is non empty trivial finite 1 -element Element of bool X
bool { the Element of X} is non empty non with_non-empty_elements finite finite-membered subset-closed V120() V252() d.union-closed Element of bool (bool { the Element of X})
bool { the Element of X} is non empty non with_non-empty_elements finite finite-membered subset-closed V120() V252() d.union-closed set
bool (bool { the Element of X}) is non empty non with_non-empty_elements finite finite-membered subset-closed V120() V252() d.union-closed set
KX is Element of bool (bool X)
TopStruct(# X,KX #) is non empty strict TopStruct
[#] TopStruct(# X,KX #) is non empty non proper Element of bool the carrier of TopStruct(# X,KX #)
the carrier of TopStruct(# X,KX #) is non empty set
bool the carrier of TopStruct(# X,KX #) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
[#] X is non empty non proper Element of bool X
SX is (X)
(SX) is Element of bool the carrier of SX
the carrier of SX is set
bool the carrier of SX is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
P is set
n is Element of the carrier of SX
Pn is Element of bool the carrier of SX
[#] SX is non proper Element of bool the carrier of SX
the_family_of SX is Element of bool (bool the carrier of SX)
bool (bool the carrier of SX) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
the topology of SX is Element of bool (bool the carrier of SX)
the non empty set is non empty set
the non empty strict non void subset-closed finite-membered finite-degree () () () () ( the non empty set ) is non empty strict non void subset-closed finite-membered finite-degree () () () () ( the non empty set )
X is non void () TopStruct
(X) is Element of bool the carrier of X
the carrier of X is set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
X is finite-membered finite-degree () () TopStruct
the carrier of X is set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool the carrier of X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
n is Element of bool (bool the carrier of X)
the topology of X is Element of bool (bool the carrier of X)
(X) is Element of bool the carrier of X
union the topology of X is Element of bool the carrier of X
bool (X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed Element of bool (bool (X))
bool (X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool (X)) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
X is finite-membered TopStruct
the carrier of X is set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool the carrier of X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
n is Element of bool (bool the carrier of X)
KX is set
SX is Element of bool the carrier of X
the topology of X is Element of bool (bool the carrier of X)
the_family_of X is Element of bool (bool the carrier of X)
X is TopStruct
(X) is Element of bool the carrier of X
the carrier of X is set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
n is TopStruct
(n) is Element of bool the carrier of n
the carrier of n is set
bool the carrier of n is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
X is TopStruct
(X) is Element of bool the carrier of X
the carrier of X is set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
the topology of X is Element of bool (bool the carrier of X)
bool (bool the carrier of X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
union the topology of X is Element of bool the carrier of X
X is TopStruct
the carrier of X is set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
n is Element of bool the carrier of X
(X) is Element of bool the carrier of X
X is TopStruct
the topology of X is Element of bool (bool the carrier of X)
the carrier of X is set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool the carrier of X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
X is TopStruct
the topology of X is Element of bool (bool the carrier of X)
the carrier of X is set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool the carrier of X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
(X) is Element of bool the carrier of X
union the topology of X is Element of bool the carrier of X
KX is set
SX is set
[:(X), the topology of X:] is Relation-like set
bool [:(X), the topology of X:] is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
KX is Relation-like (X) -defined the topology of X -valued Function-like V33((X), the topology of X) Element of bool [:(X), the topology of X:]
dom KX is Element of bool (X)
bool (X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
rng KX is Element of bool the topology of X
bool the topology of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
SX is set
{SX} is non empty trivial finite 1 -element set
KX " {SX} is Element of bool (X)
P is Element of bool the carrier of X
{P} is non empty trivial finite 1 -element Element of bool (bool the carrier of X)
KX " {P} is Element of bool (X)
n is set
KX . n is set
the_family_of X is Element of bool (bool the carrier of X)
X is TopStruct
the topology of X is Element of bool (bool the carrier of X)
the carrier of X is set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool the carrier of X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
the_family_of X is Element of bool (bool the carrier of X)
n is set
{n} is non empty trivial finite 1 -element set
({n}) is non empty non with_non-empty_elements subset-closed V252() set
bool n is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed Element of bool (bool n)
bool n is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool n) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
X is set
bool X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
n is Element of bool (bool X)
(X,n) is subset-closed V252() Element of bool (bool X)
TopStruct(# X,(X,n) #) is strict TopStruct
[#] TopStruct(# X,(X,n) #) is non proper Element of bool the carrier of TopStruct(# X,(X,n) #)
the carrier of TopStruct(# X,(X,n) #) is set
bool the carrier of TopStruct(# X,(X,n) #) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
X is set
bool X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
n is Element of bool (bool X)
(X,n) is strict (X)
(X,n) is subset-closed V252() Element of bool (bool X)
TopStruct(# X,(X,n) #) is strict TopStruct
[#] (X,n) is non proper Element of bool the carrier of (X,n)
the carrier of (X,n) is set
bool the carrier of (X,n) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
the_family_of TopStruct(# X,(X,n) #) is Element of bool (bool the carrier of TopStruct(# X,(X,n) #))
the carrier of TopStruct(# X,(X,n) #) is set
bool the carrier of TopStruct(# X,(X,n) #) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool the carrier of TopStruct(# X,(X,n) #)) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
the topology of TopStruct(# X,(X,n) #) is Element of bool (bool the carrier of TopStruct(# X,(X,n) #))
X is set
bool X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
n is non empty Element of bool (bool X)
(X,n) is strict subset-closed (X) (X)
(X,n) is non empty non with_non-empty_elements subset-closed V252() Element of bool (bool X)
TopStruct(# X,(X,n) #) is strict TopStruct
X is set
bool X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
n is finite-membered Element of bool (bool X)
(X,n) is strict subset-closed (X) (X)
(X,n) is finite-membered subset-closed V252() Element of bool (bool X)
TopStruct(# X,(X,n) #) is strict TopStruct
the_family_of (X,n) is subset-closed V252() Element of bool (bool the carrier of (X,n))
the carrier of (X,n) is set
bool the carrier of (X,n) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool the carrier of (X,n)) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
the topology of (X,n) is Element of bool (bool the carrier of (X,n))
X is set
bool X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
n is finite finite-membered Element of bool (bool X)
(X,n) is strict subset-closed finite-membered (X) (X)
(X,n) is finite-membered subset-closed V252() Element of bool (bool X)
TopStruct(# X,(X,n) #) is strict TopStruct
union n is finite Element of bool X
bool (union n) is non empty non with_non-empty_elements finite finite-membered subset-closed V120() V252() d.union-closed set
bool (bool (union n)) is non empty non with_non-empty_elements finite finite-membered subset-closed V120() V252() d.union-closed set
SX is finite finite-membered Element of bool (bool (union n))
((union n),SX) is finite finite-membered subset-closed V252() Element of bool (bool (union n))
bool (union n) is non empty non with_non-empty_elements finite finite-membered subset-closed V120() V252() d.union-closed Element of bool (bool (union n))
the carrier of (X,n) is set
the topology of (X,n) is Element of bool (bool the carrier of (X,n))
bool the carrier of (X,n) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool the carrier of (X,n)) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
union the topology of (X,n) is Element of bool the carrier of (X,n)
union (bool (union n)) is finite Element of bool (union n)
((X,n)) is Element of bool the carrier of (X,n)
X is TopStruct
the carrier of X is set
the topology of X is Element of bool (bool the carrier of X)
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
bool (bool the carrier of X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
TopStruct(# the carrier of X, the topology of X #) is strict TopStruct
( the carrier of X, the topology of X) is strict subset-closed ( the carrier of X) ( the carrier of X)
( the carrier of X, the topology of X) is subset-closed V252() Element of bool (bool the carrier of X)
TopStruct(# the carrier of X,( the carrier of X, the topology of X) #) is strict TopStruct
the_family_of X is Element of bool (bool the carrier of X)
X is non void () TopStruct
the carrier of X is set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
n 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 non with_non-empty_elements subset-closed V120() V252() d.union-closed set
{} X is empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V17() real Relation-like non-empty empty-yielding Function-like one-to-one constant functional finite finite-yielding finite-membered cardinal {} -element open integer subset-closed Function-yielding V200() V252() Element of bool the carrier of X
X is non void finite-membered TopStruct
the carrier of X is set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
the topology of X is non empty Element of bool (bool the carrier of X)
bool (bool the carrier of X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
n is set
KX is finite open Element of bool the carrier of X
X is TopStruct
the carrier of X is set
bool the carrier of X is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
- 1 is non empty ext-real non positive negative V17() real set
K58(1) is ext-real non positive V17() real integer set
the_family_of X is Element of bool (bool the carrier of X)
bool (bool the carrier of X) is non empty non with_non-empty_elements subset-closed V120() V252() d.union-closed set
the topology of X is Element of bool (bool the carrier of X)
n is ext-real non negative epsilon-transitive epsilon-connected ordinal natural V17() real finite cardinal integer set
n - 1 is ext-real V17() real set
n + (- 1) is ext-real V17() real set
K56(n,(- 1)) is ext-real V17() real set
K60(n,1) is ext-real V17() real integer set
KX is ext-real