Article WAYBEL_9, MML version 4.99.1005
:: WAYBEL_9:funcreg 1
registration
let a1 be non empty RelStr;
cluster id a1 -> Function-like quasi_total monotone;
end;
:: WAYBEL_9:attrnot 1 => WAYBEL_0:attr 5
definition
let a1, a2 be RelStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
attr a3 is antitone means
for b1, b2 being Element of the carrier of a1
st b1 <= b2
holds a3 . b2 <= a3 . b1;
end;
:: WAYBEL_9:dfs 1
definiens
let a1, a2 be non empty RelStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
a3 is antitone
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1
st b1 <= b2
holds a3 . b2 <= a3 . b1;
:: WAYBEL_9:def 1
theorem
for b1, b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is antitone(b1, b2)
iff
for b4, b5 being Element of the carrier of b1
st b4 <= b5
holds b3 . b5 <= b3 . b4;
:: WAYBEL_9:th 1
theorem
for b1, b2 being RelStr
for b3, b4 being non empty RelStr
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b6 being Function-like quasi_total Relation of the carrier of b3,the carrier of b4
st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b3,the InternalRel of b3#) &
RelStr(#the carrier of b2,the InternalRel of b2#) = RelStr(#the carrier of b4,the InternalRel of b4#) &
b5 = b6 &
b5 is monotone(b1, b2)
holds b6 is monotone(b3, b4);
:: WAYBEL_9:th 2
theorem
for b1, b2 being RelStr
for b3, b4 being non empty RelStr
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b6 being Function-like quasi_total Relation of the carrier of b3,the carrier of b4
st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b3,the InternalRel of b3#) &
RelStr(#the carrier of b2,the InternalRel of b2#) = RelStr(#the carrier of b4,the InternalRel of b4#) &
b5 = b6 &
b5 is antitone(b1, b2)
holds b6 is antitone(b3, b4);
:: WAYBEL_9:th 3
theorem
for b1, b2 being 1-sorted
for b3 being Element of bool bool the carrier of b1
for b4 being Element of bool bool the carrier of b2
st the carrier of b1 = the carrier of b2 & b3 = b4 & b3 is_a_cover_of b1
holds b4 is_a_cover_of b2;
:: WAYBEL_9:th 4
theorem
for b1 being reflexive antisymmetric with_suprema RelStr
for b2 being Element of the carrier of b1 holds
uparrow b2 = {b2} "\/" [#] b1;
:: WAYBEL_9:th 5
theorem
for b1 being reflexive antisymmetric with_infima RelStr
for b2 being Element of the carrier of b1 holds
downarrow b2 = {b2} "/\" [#] b1;
:: WAYBEL_9:th 6
theorem
for b1 being reflexive antisymmetric with_infima RelStr
for b2 being Element of the carrier of b1 holds
b2 "/\" .: uparrow b2 = {b2};
:: WAYBEL_9:th 7
theorem
for b1 being reflexive antisymmetric with_infima RelStr
for b2 being Element of the carrier of b1 holds
b2 "/\" " {b2} = uparrow b2;
:: WAYBEL_9:th 8
theorem
for b1 being non empty 1-sorted
for b2 being non empty NetStr over b1 holds
b2 is_eventually_in rng the mapping of b2;
:: WAYBEL_9:funcreg 2
registration
let a1 be non empty reflexive RelStr;
let a2 be non empty directed Element of bool the carrier of a1;
let a3 be Function-like quasi_total Relation of a2,the carrier of a1;
cluster NetStr(#a2,(the InternalRel of a1) |_2 a2,a3#) -> strict directed;
end;
:: WAYBEL_9:funcreg 3
registration
let a1 be non empty reflexive transitive RelStr;
let a2 be non empty directed Element of bool the carrier of a1;
let a3 be Function-like quasi_total Relation of a2,the carrier of a1;
cluster NetStr(#a2,(the InternalRel of a1) |_2 a2,a3#) -> transitive strict;
end;
:: WAYBEL_9:th 9
theorem
for b1 being non empty reflexive transitive RelStr
st for b2 being Element of the carrier of b1
for b3 being non empty transitive directed NetStr over b1
st b3 is eventually-directed(b1)
holds b2 "/\" sup b3 = "\/"({b2} "/\" rng netmap(b3,b1),b1)
holds b1 is satisfying_MC;
:: WAYBEL_9:th 10
theorem
for b1 being non empty RelStr
for b2 being Element of the carrier of b1
for b3 being non empty transitive directed NetStr over b1 holds
b2 "/\" b3 is non empty transitive directed NetStr over b1;
:: WAYBEL_9:funcnot 1 => WAYBEL_9:func 1
definition
let a1 be non empty RelStr;
let a2 be Element of the carrier of a1;
let a3 be non empty transitive directed NetStr over a1;
redefine func a2 "/\" a3 -> non empty transitive strict directed NetStr over a1;
end;
:: WAYBEL_9:funcreg 4
registration
let a1 be non empty RelStr;
let a2 be Element of the carrier of a1;
let a3 be non empty reflexive NetStr over a1;
cluster a2 "/\" a3 -> reflexive strict;
end;
:: WAYBEL_9:funcreg 5
registration
let a1 be non empty RelStr;
let a2 be Element of the carrier of a1;
let a3 be non empty antisymmetric NetStr over a1;
cluster a2 "/\" a3 -> antisymmetric strict;
end;
:: WAYBEL_9:funcreg 6
registration
let a1 be non empty RelStr;
let a2 be Element of the carrier of a1;
let a3 be non empty transitive NetStr over a1;
cluster a2 "/\" a3 -> transitive strict;
end;
:: WAYBEL_9:funcreg 7
registration
let a1 be non empty RelStr;
let a2 be set;
let a3 be Function-like quasi_total Relation of a2,the carrier of a1;
cluster FinSups a3 -> non empty transitive directed;
end;
:: WAYBEL_9:funcnot 2 => WAYBEL_9:func 2
definition
let a1 be non empty RelStr;
let a2 be NetStr over a1;
func inf A2 -> Element of the carrier of a1 equals
//\(the mapping of a2,a1);
end;
:: WAYBEL_9:def 2
theorem
for b1 being non empty RelStr
for b2 being NetStr over b1 holds
inf b2 = //\(the mapping of b2,b1);
:: WAYBEL_9:prednot 1 => WAYBEL_9:pred 1
definition
let a1 be RelStr;
let a2 be NetStr over a1;
pred ex_sup_of A2 means
ex_sup_of rng the mapping of a2,a1;
end;
:: WAYBEL_9:dfs 3
definiens
let a1 be RelStr;
let a2 be NetStr over a1;
To prove
ex_sup_of a2
it is sufficient to prove
thus ex_sup_of rng the mapping of a2,a1;
:: WAYBEL_9:def 3
theorem
for b1 being RelStr
for b2 being NetStr over b1 holds
ex_sup_of b2
iff
ex_sup_of rng the mapping of b2,b1;
:: WAYBEL_9:prednot 2 => WAYBEL_9:pred 2
definition
let a1 be RelStr;
let a2 be NetStr over a1;
pred ex_inf_of A2 means
ex_inf_of rng the mapping of a2,a1;
end;
:: WAYBEL_9:dfs 4
definiens
let a1 be RelStr;
let a2 be NetStr over a1;
To prove
ex_inf_of a2
it is sufficient to prove
thus ex_inf_of rng the mapping of a2,a1;
:: WAYBEL_9:def 4
theorem
for b1 being RelStr
for b2 being NetStr over b1 holds
ex_inf_of b2
iff
ex_inf_of rng the mapping of b2,b1;
:: WAYBEL_9:funcnot 3 => WAYBEL_9:func 3
definition
let a1 be RelStr;
func A1 +id -> strict NetStr over a1 means
RelStr(#the carrier of it,the InternalRel of it#) = RelStr(#the carrier of a1,the InternalRel of a1#) &
the mapping of it = id a1;
end;
:: WAYBEL_9:def 5
theorem
for b1 being RelStr
for b2 being strict NetStr over b1 holds
b2 = b1 +id
iff
RelStr(#the carrier of b2,the InternalRel of b2#) = RelStr(#the carrier of b1,the InternalRel of b1#) &
the mapping of b2 = id b1;
:: WAYBEL_9:funcreg 8
registration
let a1 be non empty RelStr;
cluster a1 +id -> non empty strict;
end;
:: WAYBEL_9:funcreg 9
registration
let a1 be reflexive RelStr;
cluster a1 +id -> reflexive strict;
end;
:: WAYBEL_9:funcreg 10
registration
let a1 be antisymmetric RelStr;
cluster a1 +id -> antisymmetric strict;
end;
:: WAYBEL_9:funcreg 11
registration
let a1 be transitive RelStr;
cluster a1 +id -> transitive strict;
end;
:: WAYBEL_9:funcreg 12
registration
let a1 be with_suprema RelStr;
cluster a1 +id -> strict directed;
end;
:: WAYBEL_9:funcreg 13
registration
let a1 be directed RelStr;
cluster a1 +id -> strict directed;
end;
:: WAYBEL_9:funcreg 14
registration
let a1 be non empty RelStr;
cluster a1 +id -> strict monotone eventually-directed;
end;
:: WAYBEL_9:funcnot 4 => WAYBEL_9:func 4
definition
let a1 be RelStr;
func A1 opp+id -> strict NetStr over a1 means
the carrier of it = the carrier of a1 & the InternalRel of it = (the InternalRel of a1) ~ & the mapping of it = id a1;
end;
:: WAYBEL_9:def 6
theorem
for b1 being RelStr
for b2 being strict NetStr over b1 holds
b2 = b1 opp+id
iff
the carrier of b2 = the carrier of b1 & the InternalRel of b2 = (the InternalRel of b1) ~ & the mapping of b2 = id b1;
:: WAYBEL_9:th 11
theorem
for b1 being RelStr holds
RelStr(#the carrier of b1 ~,the InternalRel of b1 ~#) = RelStr(#the carrier of b1 opp+id,the InternalRel of b1 opp+id#);
:: WAYBEL_9:funcreg 15
registration
let a1 be non empty RelStr;
cluster a1 opp+id -> non empty strict;
end;
:: WAYBEL_9:funcreg 16
registration
let a1 be reflexive RelStr;
cluster a1 opp+id -> reflexive strict;
end;
:: WAYBEL_9:funcreg 17
registration
let a1 be antisymmetric RelStr;
cluster a1 opp+id -> antisymmetric strict;
end;
:: WAYBEL_9:funcreg 18
registration
let a1 be transitive RelStr;
cluster a1 opp+id -> transitive strict;
end;
:: WAYBEL_9:funcreg 19
registration
let a1 be with_infima RelStr;
cluster a1 opp+id -> strict directed;
end;
:: WAYBEL_9:funcreg 20
registration
let a1 be non empty RelStr;
cluster a1 opp+id -> strict antitone eventually-filtered;
end;
:: WAYBEL_9:funcnot 5 => WAYBEL_9:func 5
definition
let a1 be non empty 1-sorted;
let a2 be non empty NetStr over a1;
let a3 be Element of the carrier of a2;
func A2 | A3 -> strict NetStr over a1 means
(for b1 being set holds
b1 in the carrier of it
iff
ex b2 being Element of the carrier of a2 st
b2 = b1 & a3 <= b2) &
the InternalRel of it = (the InternalRel of a2) |_2 the carrier of it &
the mapping of it = (the mapping of a2) | the carrier of it;
end;
:: WAYBEL_9:def 7
theorem
for b1 being non empty 1-sorted
for b2 being non empty NetStr over b1
for b3 being Element of the carrier of b2
for b4 being strict NetStr over b1 holds
b4 = b2 | b3
iff
(for b5 being set holds
b5 in the carrier of b4
iff
ex b6 being Element of the carrier of b2 st
b6 = b5 & b3 <= b6) &
the InternalRel of b4 = (the InternalRel of b2) |_2 the carrier of b4 &
the mapping of b4 = (the mapping of b2) | the carrier of b4;
:: WAYBEL_9:th 12
theorem
for b1 being non empty 1-sorted
for b2 being non empty NetStr over b1
for b3 being Element of the carrier of b2 holds
the carrier of b2 | b3 = {b4 where b4 is Element of the carrier of b2: b3 <= b4};
:: WAYBEL_9:th 13
theorem
for b1 being non empty 1-sorted
for b2 being non empty NetStr over b1
for b3 being Element of the carrier of b2 holds
the carrier of b2 | b3 c= the carrier of b2;
:: WAYBEL_9:th 14
theorem
for b1 being non empty 1-sorted
for b2 being non empty NetStr over b1
for b3 being Element of the carrier of b2 holds
b2 | b3 is full SubNetStr of b2;
:: WAYBEL_9:funcreg 21
registration
let a1 be non empty 1-sorted;
let a2 be non empty reflexive NetStr over a1;
let a3 be Element of the carrier of a2;
cluster a2 | a3 -> non empty reflexive strict;
end;
:: WAYBEL_9:funcreg 22
registration
let a1 be non empty 1-sorted;
let a2 be non empty directed NetStr over a1;
let a3 be Element of the carrier of a2;
cluster a2 | a3 -> non empty strict;
end;
:: WAYBEL_9:funcreg 23
registration
let a1 be non empty 1-sorted;
let a2 be non empty reflexive antisymmetric NetStr over a1;
let a3 be Element of the carrier of a2;
cluster a2 | a3 -> antisymmetric strict;
end;
:: WAYBEL_9:funcreg 24
registration
let a1 be non empty 1-sorted;
let a2 be non empty antisymmetric directed NetStr over a1;
let a3 be Element of the carrier of a2;
cluster a2 | a3 -> antisymmetric strict;
end;
:: WAYBEL_9:funcreg 25
registration
let a1 be non empty 1-sorted;
let a2 be non empty reflexive transitive NetStr over a1;
let a3 be Element of the carrier of a2;
cluster a2 | a3 -> transitive strict;
end;
:: WAYBEL_9:funcreg 26
registration
let a1 be non empty 1-sorted;
let a2 be non empty transitive directed NetStr over a1;
let a3 be Element of the carrier of a2;
cluster a2 | a3 -> transitive strict directed;
end;
:: WAYBEL_9:th 15
theorem
for b1 being non empty 1-sorted
for b2 being non empty reflexive NetStr over b1
for b3, b4 being Element of the carrier of b2
for b5 being Element of the carrier of b2 | b3
st b4 = b5
holds b2 . b4 = (b2 | b3) . b5;
:: WAYBEL_9:th 16
theorem
for b1 being non empty 1-sorted
for b2 being non empty directed NetStr over b1
for b3, b4 being Element of the carrier of b2
for b5 being Element of the carrier of b2 | b3
st b4 = b5
holds b2 . b4 = (b2 | b3) . b5;
:: WAYBEL_9:th 17
theorem
for b1 being non empty 1-sorted
for b2 being non empty transitive directed NetStr over b1
for b3 being Element of the carrier of b2 holds
b2 | b3 is subnet of b2;
:: WAYBEL_9:exreg 1
registration
let a1 be non empty 1-sorted;
let a2 be non empty transitive directed NetStr over a1;
cluster non empty transitive strict directed subnet of a2;
end;
:: WAYBEL_9:funcnot 6 => WAYBEL_9:func 6
definition
let a1 be non empty 1-sorted;
let a2 be non empty transitive directed NetStr over a1;
let a3 be Element of the carrier of a2;
redefine func a2 | a3 -> strict subnet of a2;
end;
:: WAYBEL_9:funcnot 7 => WAYBEL_9:func 7
definition
let a1 be non empty 1-sorted;
let a2 be 1-sorted;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
let a4 be NetStr over a1;
func A3 * A4 -> strict NetStr over a2 means
RelStr(#the carrier of it,the InternalRel of it#) = RelStr(#the carrier of a4,the InternalRel of a4#) &
the mapping of it = a3 * the mapping of a4;
end;
:: WAYBEL_9:def 8
theorem
for b1 being non empty 1-sorted
for b2 being 1-sorted
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being NetStr over b1
for b5 being strict NetStr over b2 holds
b5 = b3 * b4
iff
RelStr(#the carrier of b5,the InternalRel of b5#) = RelStr(#the carrier of b4,the InternalRel of b4#) &
the mapping of b5 = b3 * the mapping of b4;
:: WAYBEL_9:funcreg 27
registration
let a1 be non empty 1-sorted;
let a2 be 1-sorted;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
let a4 be non empty NetStr over a1;
cluster a3 * a4 -> non empty strict;
end;
:: WAYBEL_9:funcreg 28
registration
let a1 be non empty 1-sorted;
let a2 be 1-sorted;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
let a4 be reflexive NetStr over a1;
cluster a3 * a4 -> reflexive strict;
end;
:: WAYBEL_9:funcreg 29
registration
let a1 be non empty 1-sorted;
let a2 be 1-sorted;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
let a4 be antisymmetric NetStr over a1;
cluster a3 * a4 -> antisymmetric strict;
end;
:: WAYBEL_9:funcreg 30
registration
let a1 be non empty 1-sorted;
let a2 be 1-sorted;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
let a4 be transitive NetStr over a1;
cluster a3 * a4 -> transitive strict;
end;
:: WAYBEL_9:funcreg 31
registration
let a1 be non empty 1-sorted;
let a2 be 1-sorted;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
let a4 be directed NetStr over a1;
cluster a3 * a4 -> strict directed;
end;
:: WAYBEL_9:th 18
theorem
for b1 being non empty RelStr
for b2 being non empty NetStr over b1
for b3 being Element of the carrier of b1 holds
b3 "/\" * b2 = b3 "/\" b2;
:: WAYBEL_9:th 19
theorem
for b1, b2 being TopStruct
for b3 being Element of bool bool the carrier of b1
for b4 being Element of bool bool the carrier of b2
st TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#) &
b3 = b4 &
b3 is open(b1)
holds b4 is open(b2);
:: WAYBEL_9:th 20
theorem
for b1, b2 being TopStruct
for b3 being Element of bool bool the carrier of b1
for b4 being Element of bool bool the carrier of b2
st TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#) &
b3 = b4 &
b3 is closed(b1)
holds b4 is closed(b2);
:: WAYBEL_9:structnot 1 => WAYBEL_9:struct 1
definition
struct(TopStruct, RelStr) TopRelStr(#
carrier -> set,
InternalRel -> Relation of the carrier of it,the carrier of it,
topology -> Element of bool bool the carrier of it
#);
end;
:: WAYBEL_9:attrnot 2 => WAYBEL_9:attr 1
definition
let a1 be TopRelStr;
attr a1 is strict;
end;
:: WAYBEL_9:exreg 2
registration
cluster strict TopRelStr;
end;
:: WAYBEL_9:aggrnot 1 => WAYBEL_9:aggr 1
definition
let a1 be set;
let a2 be Relation of a1,a1;
let a3 be Element of bool bool a1;
aggr TopRelStr(#a1,a2,a3#) -> strict TopRelStr;
end;
:: WAYBEL_9:funcreg 32
registration
let a1 be non empty set;
let a2 be Relation of a1,a1;
let a3 be Element of bool bool a1;
cluster TopRelStr(#a1,a2,a3#) -> non empty strict;
end;
:: WAYBEL_9:funcreg 33
registration
let a1 be set;
let a2 be Relation of {a1},{a1};
let a3 be Element of bool bool {a1};
cluster TopRelStr(#{a1},a2,a3#) -> trivial strict;
end;
:: WAYBEL_9:funcreg 34
registration
let a1 be set;
let a2 be reflexive antisymmetric transitive total Relation of a1,a1;
let a3 be Element of bool bool a1;
cluster TopRelStr(#a1,a2,a3#) -> reflexive transitive antisymmetric strict;
end;
:: WAYBEL_9:exreg 3
registration
cluster non empty trivial finite discrete reflexive strict TopRelStr;
end;
:: WAYBEL_9:modenot 1
definition
mode TopLattice is TopSpace-like reflexive transitive antisymmetric with_suprema with_infima TopRelStr;
end;
:: WAYBEL_9:exreg 4
registration
cluster non empty trivial finite TopSpace-like discrete reflexive transitive antisymmetric with_suprema with_infima compact being_T2 strict TopRelStr;
end;
:: WAYBEL_9:condreg 1
registration
let a1 be non empty TopSpace-like being_T2 TopStruct;
cluster non empty -> being_T2 (SubSpace of a1);
end;
:: WAYBEL_9:th 21
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being Element of the carrier of OpenNeighborhoods b2 holds
b3 is a_neighborhood of b2;
:: WAYBEL_9:th 22
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3, b4 being Element of the carrier of OpenNeighborhoods b2 holds
b3 /\ b4 is Element of the carrier of OpenNeighborhoods b2;
:: WAYBEL_9:th 23
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3, b4 being Element of the carrier of OpenNeighborhoods b2 holds
b3 \/ b4 is Element of the carrier of OpenNeighborhoods b2;
:: WAYBEL_9:th 24
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being non empty transitive directed NetStr over b1
st b2 in Lim b3
for b4 being Element of bool the carrier of b1
st b4 = rng the mapping of b3
holds b2 in Cl b4;
:: WAYBEL_9:th 25
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima being_T2 TopRelStr
for b2 being non empty transitive directed convergent NetStr over b1
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
st b3 is continuous(b1, b1)
holds b3 . lim b2 in Lim (b3 * b2);
:: WAYBEL_9:th 26
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima being_T2 TopRelStr
for b2 being non empty transitive directed convergent NetStr over b1
for b3 being Element of the carrier of b1
st b3 "/\" is continuous(b1, b1)
holds b3 "/\" lim b2 in Lim (b3 "/\" b2);
:: WAYBEL_9:th 27
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima being_T2 TopRelStr
for b2 being Element of the carrier of b1
st for b3 being Element of the carrier of b1 holds
b3 "/\" is continuous(b1, b1)
holds uparrow b2 is closed(b1);
:: WAYBEL_9:th 28
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima compact being_T2 TopRelStr
for b2 being Element of the carrier of b1
st for b3 being Element of the carrier of b1 holds
b3 "/\" is continuous(b1, b1)
holds downarrow b2 is closed(b1);
:: WAYBEL_9:prednot 3 => WAYBEL_9:pred 3
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be non empty NetStr over a1;
let a3 be Element of the carrier of a1;
pred A3 is_a_cluster_point_of A2 means
for b1 being a_neighborhood of a3 holds
a2 is_often_in b1;
end;
:: WAYBEL_9:dfs 9
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be non empty NetStr over a1;
let a3 be Element of the carrier of a1;
To prove
a3 is_a_cluster_point_of a2
it is sufficient to prove
thus for b1 being a_neighborhood of a3 holds
a2 is_often_in b1;
:: WAYBEL_9:def 9
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty NetStr over b1
for b3 being Element of the carrier of b1 holds
b3 is_a_cluster_point_of b2
iff
for b4 being a_neighborhood of b3 holds
b2 is_often_in b4;
:: WAYBEL_9:th 29
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty transitive directed NetStr over b1
for b3 being Element of the carrier of b1
st b3 in Lim b2
holds b3 is_a_cluster_point_of b2;
:: WAYBEL_9:th 30
theorem
for b1 being non empty TopSpace-like compact being_T2 TopStruct
for b2 being non empty transitive directed NetStr over b1 holds
ex b3 being Element of the carrier of b1 st
b3 is_a_cluster_point_of b2;
:: WAYBEL_9:th 31
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty transitive directed NetStr over b1
for b3 being subnet of b2
for b4 being Element of the carrier of b1
st b4 is_a_cluster_point_of b3
holds b4 is_a_cluster_point_of b2;
:: WAYBEL_9:th 32
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty transitive directed NetStr over b1
for b3 being Element of the carrier of b1
st b3 is_a_cluster_point_of b2
holds ex b4 being subnet of b2 st
b3 in Lim b4;
:: WAYBEL_9:th 33
theorem
for b1 being non empty TopSpace-like compact being_T2 TopStruct
for b2 being non empty transitive directed NetStr over b1
st for b3, b4 being Element of the carrier of b1
st b3 is_a_cluster_point_of b2 & b4 is_a_cluster_point_of b2
holds b3 = b4
for b3 being Element of the carrier of b1
st b3 is_a_cluster_point_of b2
holds b3 in Lim b2;
:: WAYBEL_9:th 34
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being non empty transitive directed NetStr over b1
for b4 being Element of bool the carrier of b1
st b2 is_a_cluster_point_of b3 & b4 is closed(b1) & rng the mapping of b3 c= b4
holds b2 in b4;
:: WAYBEL_9:th 35
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima compact being_T2 TopRelStr
for b2 being Element of the carrier of b1
for b3 being non empty transitive directed NetStr over b1
st (for b4 being Element of the carrier of b1 holds
b4 "/\" is continuous(b1, b1)) &
b3 is eventually-directed(b1) &
b2 is_a_cluster_point_of b3
holds b2 = sup b3;
:: WAYBEL_9:th 36
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima compact being_T2 TopRelStr
for b2 being Element of the carrier of b1
for b3 being non empty transitive directed NetStr over b1
st (for b4 being Element of the carrier of b1 holds
b4 "/\" is continuous(b1, b1)) &
b3 is eventually-filtered(b1) &
b2 is_a_cluster_point_of b3
holds b2 = inf b3;
:: WAYBEL_9:th 37
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima being_T2 TopRelStr
st (for b2 being non empty transitive directed NetStr over b1
st b2 is eventually-directed(b1)
holds ex_sup_of b2 & sup b2 in Lim b2) &
(for b2 being Element of the carrier of b1 holds
b2 "/\" is continuous(b1, b1))
holds b1 is meet-continuous;
:: WAYBEL_9:th 38
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima compact being_T2 TopRelStr
st for b2 being Element of the carrier of b1 holds
b2 "/\" is continuous(b1, b1)
for b2 being non empty transitive directed NetStr over b1
st b2 is eventually-directed(b1)
holds ex_sup_of b2 & sup b2 in Lim b2;
:: WAYBEL_9:th 39
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima compact being_T2 TopRelStr
st for b2 being Element of the carrier of b1 holds
b2 "/\" is continuous(b1, b1)
for b2 being non empty transitive directed NetStr over b1
st b2 is eventually-filtered(b1)
holds ex_inf_of b2 & inf b2 in Lim b2;
:: WAYBEL_9:th 40
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima compact being_T2 TopRelStr
st for b2 being Element of the carrier of b1 holds
b2 "/\" is continuous(b1, b1)
holds b1 is bounded;
:: WAYBEL_9:th 41
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima compact being_T2 TopRelStr
st for b2 being Element of the carrier of b1 holds
b2 "/\" is continuous(b1, b1)
holds b1 is meet-continuous;