Article SETLIM_1, MML version 4.99.1005
:: SETLIM_1:th 1
theorem
for b1, b2 being set
st b1 <> {} &
(for b3 being set
st b3 in b1
holds b3 = b2)
holds union b1 = b2;
:: SETLIM_1:th 2
theorem
for b1, b2 being set
st b1 <> {} &
(for b3 being set
st b3 in b1
holds b3 = b2)
holds meet b1 = b2;
:: SETLIM_1:th 3
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,b1
for b3 being Element of NAT holds
{b2 . b4 where b4 is Element of NAT: b3 <= b4} <> {};
:: SETLIM_1:th 4
theorem
for b1, b2 being Element of NAT
for b3 being set
for b4 being Function-like quasi_total Relation of NAT,b3 holds
b4 . (b1 + b2) in {b4 . b5 where b5 is Element of NAT: b1 <= b5};
:: SETLIM_1:th 5
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Function-like quasi_total Relation of NAT,b2 holds
{b3 . b4 where b4 is Element of NAT: b1 <= b4} = {b3 . b4 where b4 is Element of NAT: b1 + 1 <= b4} \/ {b3 . b1};
:: SETLIM_1:th 6
theorem
for b1 being Element of NAT
for b2, b3 being set
for b4 being Function-like quasi_total Relation of NAT,b2 holds
for b5 being Element of NAT holds
b3 in b4 . (b1 + b5)
iff
for b5 being set
st b5 in {b4 . b6 where b6 is Element of NAT: b1 <= b6}
holds b3 in b5;
:: SETLIM_1:th 7
theorem
for b1 being set
for b2 being non empty set
for b3 being Function-like quasi_total Relation of NAT,b2 holds
b1 in rng b3
iff
ex b4 being Element of NAT st
b1 = b3 . b4;
:: SETLIM_1:th 8
theorem
for b1, b2 being non empty set
for b3 being Function-like quasi_total Relation of b1,b2 holds
rng b3 <> {} & b3 <> {};
:: SETLIM_1:th 9
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of NAT,b1 holds
rng b2 = {b2 . b3 where b3 is Element of NAT: 0 <= b3};
:: SETLIM_1:th 10
theorem
for b1 being Element of NAT
for b2 being non empty set
for b3 being Function-like quasi_total Relation of NAT,b2 holds
rng (b3 ^\ b1) = {b3 . b4 where b4 is Element of NAT: b1 <= b4};
:: SETLIM_1:th 12
theorem
for b1, b2 being set
for b3 being Function-like quasi_total Relation of NAT,bool b1 holds
b2 in meet rng b3
iff
for b4 being Element of NAT holds
b2 in b3 . b4;
:: SETLIM_1:th 13
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
Intersection b2 = meet rng b2;
:: SETLIM_1:th 14
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
Intersection b2 c= Union b2;
:: SETLIM_1:th 15
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1
st for b4 being Element of NAT holds
b3 . b4 = b2
holds Union b3 = b2;
:: SETLIM_1:th 16
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1
st for b4 being Element of NAT holds
b3 . b4 = b2
holds Intersection b3 = b2;
:: SETLIM_1:th 17
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
st b2 is constant
holds Union b2 = Intersection b2;
:: SETLIM_1:th 18
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1
st b3 is constant & the_value_of b3 = b2
for b4 being Element of NAT holds
union {b3 . b5 where b5 is Element of NAT: b4 <= b5} = b2;
:: SETLIM_1:th 19
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1
st b3 is constant & the_value_of b3 = b2
for b4 being Element of NAT holds
meet {b3 . b5 where b5 is Element of NAT: b4 <= b5} = b2;
:: SETLIM_1:th 20
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
ex b3 being Function-like quasi_total Relation of NAT,bool b1 st
for b4 being Element of NAT
for b5 being set
st b5 = {b2 . b6 where b6 is Element of NAT: b4 <= b6}
holds b3 . b4 = meet b5;
:: SETLIM_1:th 21
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
ex b3 being Function-like quasi_total Relation of NAT,bool b1 st
for b4 being Element of NAT
for b5 being set
st b5 = {b2 . b6 where b6 is Element of NAT: b4 <= b6}
holds b3 . b4 = union b5;
:: SETLIM_1:attrnot 1 => SETLIM_1:attr 1
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
attr a2 is monotone means
(a2 is not non-decreasing(a1)) implies a2 is non-increasing(a1);
end;
:: SETLIM_1:dfs 1
definiens
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
To prove
a2 is monotone
it is sufficient to prove
thus (a2 is not non-decreasing(a1)) implies a2 is non-increasing(a1);
:: SETLIM_1:def 1
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
b2 is monotone(b1)
iff
(b2 is non-decreasing(b1) or b2 is non-increasing(b1));
:: SETLIM_1:funcnot 1 => SETLIM_1:func 1
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
func inferior_setsequence A2 -> Function-like quasi_total Relation of NAT,bool a1 means
for b1 being Element of NAT
for b2 being set
st b2 = {a2 . b3 where b3 is Element of NAT: b1 <= b3}
holds it . b1 = meet b2;
end;
:: SETLIM_1:def 2
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1 holds
b3 = inferior_setsequence b2
iff
for b4 being Element of NAT
for b5 being set
st b5 = {b2 . b6 where b6 is Element of NAT: b4 <= b6}
holds b3 . b4 = meet b5;
:: SETLIM_1:funcnot 2 => SETLIM_1:func 2
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
func superior_setsequence A2 -> Function-like quasi_total Relation of NAT,bool a1 means
for b1 being Element of NAT
for b2 being set
st b2 = {a2 . b3 where b3 is Element of NAT: b1 <= b3}
holds it . b1 = union b2;
end;
:: SETLIM_1:def 3
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1 holds
b3 = superior_setsequence b2
iff
for b4 being Element of NAT
for b5 being set
st b5 = {b2 . b6 where b6 is Element of NAT: b4 <= b6}
holds b3 . b4 = union b5;
:: SETLIM_1:th 22
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
(inferior_setsequence b2) . 0 = Intersection b2;
:: SETLIM_1:th 23
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
(superior_setsequence b2) . 0 = Union b2;
:: SETLIM_1:th 24
theorem
for b1 being Element of NAT
for b2, b3 being set
for b4 being Function-like quasi_total Relation of NAT,bool b2 holds
b3 in (inferior_setsequence b4) . b1
iff
for b5 being Element of NAT holds
b3 in b4 . (b1 + b5);
:: SETLIM_1:th 25
theorem
for b1 being Element of NAT
for b2, b3 being set
for b4 being Function-like quasi_total Relation of NAT,bool b2 holds
b3 in (superior_setsequence b4) . b1
iff
ex b5 being Element of NAT st
b3 in b4 . (b1 + b5);
:: SETLIM_1:th 26
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Function-like quasi_total Relation of NAT,bool b2 holds
(inferior_setsequence b3) . b1 = ((inferior_setsequence b3) . (b1 + 1)) /\ (b3 . b1);
:: SETLIM_1:th 27
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Function-like quasi_total Relation of NAT,bool b2 holds
(superior_setsequence b3) . b1 = ((superior_setsequence b3) . (b1 + 1)) \/ (b3 . b1);
:: SETLIM_1:th 28
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
inferior_setsequence b2 is non-decreasing(b1);
:: SETLIM_1:th 29
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
superior_setsequence b2 is non-increasing(b1);
:: SETLIM_1:th 30
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
inferior_setsequence b2 is monotone(b1) & superior_setsequence b2 is monotone(b1);
:: SETLIM_1:funcreg 1
registration
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
cluster inferior_setsequence a2 -> Function-like quasi_total non-decreasing;
end;
:: SETLIM_1:funcreg 2
registration
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
cluster superior_setsequence a2 -> Function-like quasi_total non-increasing;
end;
:: SETLIM_1:th 31
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Function-like quasi_total Relation of NAT,bool b2 holds
Intersection b3 c= (inferior_setsequence b3) . b1;
:: SETLIM_1:th 32
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Function-like quasi_total Relation of NAT,bool b2 holds
(superior_setsequence b3) . b1 c= Union b3;
:: SETLIM_1:th 33
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
for b3 being Element of NAT holds
{b2 . b4 where b4 is Element of NAT: b3 <= b4} is Element of bool bool b1;
:: SETLIM_1:th 35
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
Union b2 = (Intersection Complement b2) `;
:: SETLIM_1:th 36
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Function-like quasi_total Relation of NAT,bool b2 holds
(inferior_setsequence b3) . b1 = ((superior_setsequence Complement b3) . b1) `;
:: SETLIM_1:th 37
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Function-like quasi_total Relation of NAT,bool b2 holds
(superior_setsequence b3) . b1 = ((inferior_setsequence Complement b3) . b1) `;
:: SETLIM_1:th 38
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
Complement inferior_setsequence b2 = superior_setsequence Complement b2;
:: SETLIM_1:th 39
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
Complement superior_setsequence b2 = inferior_setsequence Complement b2;
:: SETLIM_1:th 40
theorem
for b1 being set
for b2, b3, b4 being Function-like quasi_total Relation of NAT,bool b1
st for b5 being Element of NAT holds
b2 . b5 = (b3 . b5) \/ (b4 . b5)
for b5 being Element of NAT holds
((inferior_setsequence b3) . b5) \/ ((inferior_setsequence b4) . b5) c= (inferior_setsequence b2) . b5;
:: SETLIM_1:th 41
theorem
for b1 being set
for b2, b3, b4 being Function-like quasi_total Relation of NAT,bool b1
st for b5 being Element of NAT holds
b2 . b5 = (b3 . b5) /\ (b4 . b5)
for b5 being Element of NAT holds
(inferior_setsequence b2) . b5 = ((inferior_setsequence b3) . b5) /\ ((inferior_setsequence b4) . b5);
:: SETLIM_1:th 42
theorem
for b1 being set
for b2, b3, b4 being Function-like quasi_total Relation of NAT,bool b1
st for b5 being Element of NAT holds
b2 . b5 = (b3 . b5) \/ (b4 . b5)
for b5 being Element of NAT holds
(superior_setsequence b2) . b5 = ((superior_setsequence b3) . b5) \/ ((superior_setsequence b4) . b5);
:: SETLIM_1:th 43
theorem
for b1 being set
for b2, b3, b4 being Function-like quasi_total Relation of NAT,bool b1
st for b5 being Element of NAT holds
b2 . b5 = (b3 . b5) /\ (b4 . b5)
for b5 being Element of NAT holds
(superior_setsequence b2) . b5 c= ((superior_setsequence b3) . b5) /\ ((superior_setsequence b4) . b5);
:: SETLIM_1:th 44
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1
st b3 is constant & the_value_of b3 = b2
for b4 being Element of NAT holds
(inferior_setsequence b3) . b4 = b2;
:: SETLIM_1:th 45
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1
st b3 is constant & the_value_of b3 = b2
for b4 being Element of NAT holds
(superior_setsequence b3) . b4 = b2;
:: SETLIM_1:th 46
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Function-like quasi_total Relation of NAT,bool b2
st b3 is non-decreasing(b2)
holds b3 . b1 c= (superior_setsequence b3) . (b1 + 1);
:: SETLIM_1:th 47
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Function-like quasi_total Relation of NAT,bool b2
st b3 is non-decreasing(b2)
holds (superior_setsequence b3) . b1 = (superior_setsequence b3) . (b1 + 1);
:: SETLIM_1:th 48
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Function-like quasi_total Relation of NAT,bool b2
st b3 is non-decreasing(b2)
holds (superior_setsequence b3) . b1 = Union b3;
:: SETLIM_1:th 49
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
st b2 is non-decreasing(b1)
holds Intersection superior_setsequence b2 = Union b2;
:: SETLIM_1:th 50
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Function-like quasi_total Relation of NAT,bool b2
st b3 is non-decreasing(b2)
holds b3 . b1 c= (inferior_setsequence b3) . (b1 + 1);
:: SETLIM_1:th 51
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Function-like quasi_total Relation of NAT,bool b2
st b3 is non-decreasing(b2)
holds (inferior_setsequence b3) . b1 = b3 . b1;
:: SETLIM_1:th 52
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
st b2 is non-decreasing(b1)
holds inferior_setsequence b2 = b2;
:: SETLIM_1:th 53
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Function-like quasi_total Relation of NAT,bool b2
st b3 is non-increasing(b2)
holds (superior_setsequence b3) . (b1 + 1) c= b3 . b1;
:: SETLIM_1:th 54
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Function-like quasi_total Relation of NAT,bool b2
st b3 is non-increasing(b2)
holds (superior_setsequence b3) . b1 = b3 . b1;
:: SETLIM_1:th 55
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
st b2 is non-increasing(b1)
holds superior_setsequence b2 = b2;
:: SETLIM_1:th 56
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Function-like quasi_total Relation of NAT,bool b2
st b3 is non-increasing(b2)
holds (inferior_setsequence b3) . (b1 + 1) c= b3 . b1;
:: SETLIM_1:th 57
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Function-like quasi_total Relation of NAT,bool b2
st b3 is non-increasing(b2)
holds (inferior_setsequence b3) . b1 = (inferior_setsequence b3) . (b1 + 1);
:: SETLIM_1:th 58
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Function-like quasi_total Relation of NAT,bool b2
st b3 is non-increasing(b2)
holds (inferior_setsequence b3) . b1 = Intersection b3;
:: SETLIM_1:th 59
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
st b2 is non-increasing(b1)
holds Union inferior_setsequence b2 = Intersection b2;
:: SETLIM_1:funcnot 3 => KURATO_2:func 4
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
func lim_inf A2 -> Element of bool a1 equals
Union inferior_setsequence a2;
end;
:: SETLIM_1:def 4
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
lim_inf b2 = Union inferior_setsequence b2;
:: SETLIM_1:funcnot 4 => KURATO_2:func 5
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
func lim_sup A2 -> Element of bool a1 equals
Intersection superior_setsequence a2;
end;
:: SETLIM_1:def 5
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
lim_sup b2 = Intersection superior_setsequence b2;
:: SETLIM_1:funcnot 5 => SETLIM_1:func 3
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
assume a2 is convergent(a1);
func lim A2 -> Element of bool a1 means
it = lim_sup a2 & it = lim_inf a2;
end;
:: SETLIM_1:def 6
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
st b2 is convergent(b1)
for b3 being Element of bool b1 holds
b3 = lim b2
iff
b3 = lim_sup b2 & b3 = lim_inf b2;
:: SETLIM_1:th 60
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
Intersection b2 c= lim_inf b2;
:: SETLIM_1:th 61
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
lim_inf b2 = lim inferior_setsequence b2;
:: SETLIM_1:th 62
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
lim_sup b2 = lim superior_setsequence b2;
:: SETLIM_1:th 63
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
lim_sup b2 = (lim_inf Complement b2) `;
:: SETLIM_1:th 64
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1
st b3 is constant & the_value_of b3 = b2
holds b3 is convergent(b1) & lim b3 = b2 & lim_inf b3 = b2 & lim_sup b3 = b2;
:: SETLIM_1:th 65
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
st b2 is non-decreasing(b1)
holds lim_sup b2 = Union b2;
:: SETLIM_1:th 66
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
st b2 is non-decreasing(b1)
holds lim_inf b2 = Union b2;
:: SETLIM_1:th 67
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
st b2 is non-increasing(b1)
holds lim_sup b2 = Intersection b2;
:: SETLIM_1:th 68
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
st b2 is non-increasing(b1)
holds lim_inf b2 = Intersection b2;
:: SETLIM_1:th 69
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
st b2 is non-decreasing(b1)
holds b2 is convergent(b1) & lim b2 = Union b2;
:: SETLIM_1:th 70
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
st b2 is non-increasing(b1)
holds b2 is convergent(b1) & lim b2 = Intersection b2;
:: SETLIM_1:th 71
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
st b2 is monotone(b1)
holds b2 is convergent(b1);
:: SETLIM_1:attrnot 2 => SETLIM_1:attr 2
definition
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be SetSequence of a2;
redefine attr a3 is constant means
ex b1 being Element of a2 st
for b2 being Element of NAT holds
a3 . b2 = b1;
end;
:: SETLIM_1:dfs 7
definiens
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be SetSequence of a2;
To prove
a1 is constant
it is sufficient to prove
thus ex b1 being Element of a2 st
for b2 being Element of NAT holds
a3 . b2 = b1;
:: SETLIM_1:def 7
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2 holds
b3 is constant
iff
ex b4 being Element of b2 st
for b5 being Element of NAT holds
b3 . b5 = b4;
:: SETLIM_1:funcnot 6 => SETLIM_1:func 4
definition
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be SetSequence of a2;
func @inferior_setsequence A3 -> SetSequence of a2 equals
inferior_setsequence a3;
end;
:: SETLIM_1:def 8
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2 holds
@inferior_setsequence b3 = inferior_setsequence b3;
:: SETLIM_1:funcnot 7 => SETLIM_1:func 5
definition
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be SetSequence of a2;
func @superior_setsequence A3 -> SetSequence of a2 equals
superior_setsequence a3;
end;
:: SETLIM_1:def 9
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2 holds
@superior_setsequence b3 = superior_setsequence b3;
:: SETLIM_1:th 72
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2 holds
@superior_setsequence b3 is non-increasing(b1);
:: SETLIM_1:th 73
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2 holds
@inferior_setsequence b3 is non-decreasing(b1);
:: SETLIM_1:funcreg 3
registration
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be SetSequence of a2;
cluster @superior_setsequence a3 -> non-increasing;
end;
:: SETLIM_1:funcreg 4
registration
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be SetSequence of a2;
cluster @inferior_setsequence a3 -> non-decreasing;
end;
:: SETLIM_1:funcnot 8 => SETLIM_1:func 6
definition
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be SetSequence of a2;
func lim_inf A3 -> Element of a2 equals
Union @inferior_setsequence a3;
end;
:: SETLIM_1:def 10
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2 holds
lim_inf b3 = Union @inferior_setsequence b3;
:: SETLIM_1:funcnot 9 => SETLIM_1:func 7
definition
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be SetSequence of a2;
func lim_sup A3 -> Element of a2 equals
Intersection @superior_setsequence a3;
end;
:: SETLIM_1:def 11
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2 holds
lim_sup b3 = Intersection @superior_setsequence b3;
:: SETLIM_1:attrnot 3 => SETLIM_1:attr 3
definition
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be SetSequence of a2;
attr a3 is convergent means
lim_sup a3 = lim_inf a3;
end;
:: SETLIM_1:dfs 12
definiens
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be SetSequence of a2;
To prove
a3 is convergent
it is sufficient to prove
thus lim_sup a3 = lim_inf a3;
:: SETLIM_1:def 12
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2 holds
b3 is convergent(b1, b2)
iff
lim_sup b3 = lim_inf b3;
:: SETLIM_1:funcnot 10 => SETLIM_1:func 8
definition
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be SetSequence of a2;
assume a3 is convergent(a1, a2);
func lim A3 -> Element of a2 means
it = lim_sup a3 & it = lim_inf a3;
end;
:: SETLIM_1:def 13
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2
st b3 is convergent(b1, b2)
for b4 being Element of b2 holds
b4 = lim b3
iff
b4 = lim_sup b3 & b4 = lim_inf b3;
:: SETLIM_1:th 74
theorem
for b1, b2 being set
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b4 being SetSequence of b3 holds
b2 in lim_sup b4
iff
for b5 being Element of NAT holds
ex b6 being Element of NAT st
b2 in b4 . (b5 + b6);
:: SETLIM_1:th 75
theorem
for b1, b2 being set
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b4 being SetSequence of b3 holds
b2 in lim_inf b4
iff
ex b5 being Element of NAT st
for b6 being Element of NAT holds
b2 in b4 . (b5 + b6);
:: SETLIM_1:th 76
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2 holds
Intersection b3 c= lim_inf b3;
:: SETLIM_1:th 77
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2 holds
lim_sup b3 c= Union b3;
:: SETLIM_1:th 78
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2 holds
lim_inf b3 c= lim_sup b3;
:: SETLIM_1:funcnot 11 => SETLIM_1:func 9
definition
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be SetSequence of a2;
func @Complement A3 -> SetSequence of a2 equals
Complement a3;
end;
:: SETLIM_1:def 14
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2 holds
@Complement b3 = Complement b3;
:: SETLIM_1:th 79
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2 holds
lim_inf b3 = (lim_sup @Complement b3) `;
:: SETLIM_1:th 80
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2 holds
lim_sup b3 = (lim_inf @Complement b3) `;
:: SETLIM_1:th 81
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4, b5 being SetSequence of b2
st for b6 being Element of NAT holds
b3 . b6 = (b4 . b6) \/ (b5 . b6)
holds (lim_inf b4) \/ lim_inf b5 c= lim_inf b3;
:: SETLIM_1:th 82
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4, b5 being SetSequence of b2
st for b6 being Element of NAT holds
b3 . b6 = (b4 . b6) /\ (b5 . b6)
holds lim_inf b3 = (lim_inf b4) /\ lim_inf b5;
:: SETLIM_1:th 83
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4, b5 being SetSequence of b2
st for b6 being Element of NAT holds
b3 . b6 = (b4 . b6) \/ (b5 . b6)
holds lim_sup b3 = (lim_sup b4) \/ lim_sup b5;
:: SETLIM_1:th 84
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4, b5 being SetSequence of b2
st for b6 being Element of NAT holds
b3 . b6 = (b4 . b6) /\ (b5 . b6)
holds lim_sup b3 c= (lim_sup b4) /\ lim_sup b5;
:: SETLIM_1:th 85
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b4 being SetSequence of b3
st b4 is constant & the_value_of b4 = b2
holds b4 is convergent(b1, b3) & lim b4 = b2 & lim_inf b4 = b2 & lim_sup b4 = b2;
:: SETLIM_1:th 86
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2
st b3 is non-decreasing(b1)
holds lim_sup b3 = Union b3;
:: SETLIM_1:th 87
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2
st b3 is non-decreasing(b1)
holds lim_inf b3 = Union b3;
:: SETLIM_1:th 88
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2
st b3 is non-decreasing(b1)
holds b3 is convergent(b1, b2) & lim b3 = Union b3;
:: SETLIM_1:th 89
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2
st b3 is non-increasing(b1)
holds lim_sup b3 = Intersection b3;
:: SETLIM_1:th 90
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2
st b3 is non-increasing(b1)
holds lim_inf b3 = Intersection b3;
:: SETLIM_1:th 91
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2
st b3 is non-increasing(b1)
holds b3 is convergent(b1, b2) & lim b3 = Intersection b3;
:: SETLIM_1:th 92
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being SetSequence of b2
st b3 is monotone(b1)
holds b3 is convergent(b1, b2);