Article SETLIM_2, MML version 4.99.1005
:: SETLIM_2:th 1
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 = Intersection (b3 ^\ b1);
:: SETLIM_2:th 2
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 = Union (b3 ^\ b1);
:: SETLIM_2:funcnot 1 => SETLIM_2:func 1
definition
let a1 be set;
let a2, a3 be Function-like quasi_total Relation of NAT,bool a1;
func A2 (/\) A3 -> Function-like quasi_total Relation of NAT,bool a1 means
for b1 being Element of NAT holds
it . b1 = (a2 . b1) /\ (a3 . b1);
commutativity;
:: for a1 being set
:: for a2, a3 being Function-like quasi_total Relation of NAT,bool a1 holds
:: a2 (/\) a3 = a3 (/\) a2;
end;
:: SETLIM_2:def 1
theorem
for b1 being set
for b2, b3, b4 being Function-like quasi_total Relation of NAT,bool b1 holds
b4 = b2 (/\) b3
iff
for b5 being Element of NAT holds
b4 . b5 = (b2 . b5) /\ (b3 . b5);
:: SETLIM_2:funcnot 2 => SETLIM_2:func 2
definition
let a1 be set;
let a2, a3 be Function-like quasi_total Relation of NAT,bool a1;
func A2 (\/) A3 -> Function-like quasi_total Relation of NAT,bool a1 means
for b1 being Element of NAT holds
it . b1 = (a2 . b1) \/ (a3 . b1);
commutativity;
:: for a1 being set
:: for a2, a3 being Function-like quasi_total Relation of NAT,bool a1 holds
:: a2 (\/) a3 = a3 (\/) a2;
end;
:: SETLIM_2:def 2
theorem
for b1 being set
for b2, b3, b4 being Function-like quasi_total Relation of NAT,bool b1 holds
b4 = b2 (\/) b3
iff
for b5 being Element of NAT holds
b4 . b5 = (b2 . b5) \/ (b3 . b5);
:: SETLIM_2:funcnot 3 => SETLIM_2:func 3
definition
let a1 be set;
let a2, a3 be Function-like quasi_total Relation of NAT,bool a1;
func A2 (\) A3 -> Function-like quasi_total Relation of NAT,bool a1 means
for b1 being Element of NAT holds
it . b1 = (a2 . b1) \ (a3 . b1);
end;
:: SETLIM_2:def 3
theorem
for b1 being set
for b2, b3, b4 being Function-like quasi_total Relation of NAT,bool b1 holds
b4 = b2 (\) b3
iff
for b5 being Element of NAT holds
b4 . b5 = (b2 . b5) \ (b3 . b5);
:: SETLIM_2:funcnot 4 => SETLIM_2:func 4
definition
let a1 be set;
let a2, a3 be Function-like quasi_total Relation of NAT,bool a1;
func A2 (\+\) A3 -> Function-like quasi_total Relation of NAT,bool a1 means
for b1 being Element of NAT holds
it . b1 = (a2 . b1) \+\ (a3 . b1);
commutativity;
:: for a1 being set
:: for a2, a3 being Function-like quasi_total Relation of NAT,bool a1 holds
:: a2 (\+\) a3 = a3 (\+\) a2;
end;
:: SETLIM_2:def 4
theorem
for b1 being set
for b2, b3, b4 being Function-like quasi_total Relation of NAT,bool b1 holds
b4 = b2 (\+\) b3
iff
for b5 being Element of NAT holds
b4 . b5 = (b2 . b5) \+\ (b3 . b5);
:: SETLIM_2:th 3
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1 holds
b2 (\+\) b3 = (b2 (\) b3) (\/) (b3 (\) b2);
:: SETLIM_2:th 4
theorem
for b1 being Element of NAT
for b2 being set
for b3, b4 being Function-like quasi_total Relation of NAT,bool b2 holds
(b3 (/\) b4) ^\ b1 = (b3 ^\ b1) (/\) (b4 ^\ b1);
:: SETLIM_2:th 5
theorem
for b1 being Element of NAT
for b2 being set
for b3, b4 being Function-like quasi_total Relation of NAT,bool b2 holds
(b3 (\/) b4) ^\ b1 = (b3 ^\ b1) (\/) (b4 ^\ b1);
:: SETLIM_2:th 6
theorem
for b1 being Element of NAT
for b2 being set
for b3, b4 being Function-like quasi_total Relation of NAT,bool b2 holds
(b3 (\) b4) ^\ b1 = (b3 ^\ b1) (\) (b4 ^\ b1);
:: SETLIM_2:th 7
theorem
for b1 being Element of NAT
for b2 being set
for b3, b4 being Function-like quasi_total Relation of NAT,bool b2 holds
(b3 (\+\) b4) ^\ b1 = (b3 ^\ b1) (\+\) (b4 ^\ b1);
:: SETLIM_2:th 8
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1 holds
Union (b2 (/\) b3) c= (Union b2) /\ Union b3;
:: SETLIM_2:th 9
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1 holds
Union (b2 (\/) b3) = (Union b2) \/ Union b3;
:: SETLIM_2:th 10
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1 holds
(Union b2) \ Union b3 c= Union (b2 (\) b3);
:: SETLIM_2:th 11
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1 holds
(Union b2) \+\ Union b3 c= Union (b2 (\+\) b3);
:: SETLIM_2:th 12
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1 holds
Intersection (b2 (/\) b3) = (Intersection b2) /\ Intersection b3;
:: SETLIM_2:th 13
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1 holds
(Intersection b2) \/ Intersection b3 c= Intersection (b2 (\/) b3);
:: SETLIM_2:th 14
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1 holds
Intersection (b2 (\) b3) c= (Intersection b2) \ Intersection b3;
:: SETLIM_2:funcnot 5 => SETLIM_2:func 5
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
let a3 be Element of bool a1;
func A3 (/\) A2 -> Function-like quasi_total Relation of NAT,bool a1 means
for b1 being Element of NAT holds
it . b1 = a3 /\ (a2 . b1);
end;
:: SETLIM_2:def 5
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
for b3 being Element of bool b1
for b4 being Function-like quasi_total Relation of NAT,bool b1 holds
b4 = b3 (/\) b2
iff
for b5 being Element of NAT holds
b4 . b5 = b3 /\ (b2 . b5);
:: SETLIM_2:funcnot 6 => SETLIM_2:func 6
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
let a3 be Element of bool a1;
func A3 (\/) A2 -> Function-like quasi_total Relation of NAT,bool a1 means
for b1 being Element of NAT holds
it . b1 = a3 \/ (a2 . b1);
end;
:: SETLIM_2:def 6
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
for b3 being Element of bool b1
for b4 being Function-like quasi_total Relation of NAT,bool b1 holds
b4 = b3 (\/) b2
iff
for b5 being Element of NAT holds
b4 . b5 = b3 \/ (b2 . b5);
:: SETLIM_2:funcnot 7 => SETLIM_2:func 7
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
let a3 be Element of bool a1;
func A3 (\) A2 -> Function-like quasi_total Relation of NAT,bool a1 means
for b1 being Element of NAT holds
it . b1 = a3 \ (a2 . b1);
end;
:: SETLIM_2:def 7
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
for b3 being Element of bool b1
for b4 being Function-like quasi_total Relation of NAT,bool b1 holds
b4 = b3 (\) b2
iff
for b5 being Element of NAT holds
b4 . b5 = b3 \ (b2 . b5);
:: SETLIM_2:funcnot 8 => SETLIM_2:func 8
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
let a3 be Element of bool a1;
func A2 (\) A3 -> Function-like quasi_total Relation of NAT,bool a1 means
for b1 being Element of NAT holds
it . b1 = (a2 . b1) \ a3;
end;
:: SETLIM_2:def 8
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
for b3 being Element of bool b1
for b4 being Function-like quasi_total Relation of NAT,bool b1 holds
b4 = b2 (\) b3
iff
for b5 being Element of NAT holds
b4 . b5 = (b2 . b5) \ b3;
:: SETLIM_2:funcnot 9 => SETLIM_2:func 9
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
let a3 be Element of bool a1;
func A3 (\+\) A2 -> Function-like quasi_total Relation of NAT,bool a1 means
for b1 being Element of NAT holds
it . b1 = a3 \+\ (a2 . b1);
end;
:: SETLIM_2:def 9
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
for b3 being Element of bool b1
for b4 being Function-like quasi_total Relation of NAT,bool b1 holds
b4 = b3 (\+\) b2
iff
for b5 being Element of NAT holds
b4 . b5 = b3 \+\ (b2 . b5);
:: SETLIM_2: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 holds
b2 (\+\) b3 = (b2 (\) b3) (\/) (b3 (\) b2);
:: SETLIM_2:th 16
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Element of bool b2
for b4 being Function-like quasi_total Relation of NAT,bool b2 holds
(b3 (/\) b4) ^\ b1 = b3 (/\) (b4 ^\ b1);
:: SETLIM_2:th 17
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Element of bool b2
for b4 being Function-like quasi_total Relation of NAT,bool b2 holds
(b3 (\/) b4) ^\ b1 = b3 (\/) (b4 ^\ b1);
:: SETLIM_2:th 18
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Element of bool b2
for b4 being Function-like quasi_total Relation of NAT,bool b2 holds
(b3 (\) b4) ^\ b1 = b3 (\) (b4 ^\ b1);
:: SETLIM_2:th 19
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Element of bool b2
for b4 being Function-like quasi_total Relation of NAT,bool b2 holds
(b4 (\) b3) ^\ b1 = (b4 ^\ b1) (\) b3;
:: SETLIM_2:th 20
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Element of bool b2
for b4 being Function-like quasi_total Relation of NAT,bool b2 holds
(b3 (\+\) b4) ^\ b1 = b3 (\+\) (b4 ^\ b1);
:: SETLIM_2:th 21
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 non-increasing(b1)
holds b2 (/\) b3 is non-increasing(b1);
:: SETLIM_2:th 22
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 non-decreasing(b1)
holds b2 (/\) b3 is non-decreasing(b1);
:: SETLIM_2:th 23
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 monotone(b1)
holds b2 (/\) b3 is monotone(b1);
:: SETLIM_2:th 24
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 non-increasing(b1)
holds b2 (\/) b3 is non-increasing(b1);
:: SETLIM_2:th 25
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 non-decreasing(b1)
holds b2 (\/) b3 is non-decreasing(b1);
:: SETLIM_2:th 26
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 monotone(b1)
holds b2 (\/) b3 is monotone(b1);
:: SETLIM_2:th 27
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 non-increasing(b1)
holds b2 (\) b3 is non-decreasing(b1);
:: SETLIM_2:th 28
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 non-decreasing(b1)
holds b2 (\) b3 is non-increasing(b1);
:: SETLIM_2:th 29
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 monotone(b1)
holds b2 (\) b3 is monotone(b1);
:: SETLIM_2:th 30
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 non-increasing(b1)
holds b3 (\) b2 is non-increasing(b1);
:: SETLIM_2:th 31
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 non-decreasing(b1)
holds b3 (\) b2 is non-decreasing(b1);
:: SETLIM_2:th 32
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 monotone(b1)
holds b3 (\) b2 is monotone(b1);
:: SETLIM_2:th 33
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1 holds
Intersection (b2 (/\) b3) = b2 /\ Intersection b3;
:: SETLIM_2:th 34
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1 holds
Intersection (b2 (\/) b3) = b2 \/ Intersection b3;
:: SETLIM_2:th 35
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1 holds
Intersection (b2 (\) b3) c= b2 \ Intersection b3;
:: SETLIM_2:th 36
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1 holds
Intersection (b3 (\) b2) = (Intersection b3) \ b2;
:: SETLIM_2:th 37
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1 holds
Intersection (b2 (\+\) b3) c= b2 \+\ Intersection b3;
:: SETLIM_2:th 38
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1 holds
Union (b2 (/\) b3) = b2 /\ Union b3;
:: SETLIM_2:th 39
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1 holds
Union (b2 (\/) b3) = b2 \/ Union b3;
:: SETLIM_2:th 40
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1 holds
b2 \ Union b3 c= Union (b2 (\) b3);
:: SETLIM_2:th 41
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1 holds
Union (b3 (\) b2) = (Union b3) \ b2;
:: SETLIM_2:th 42
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1 holds
b2 \+\ Union b3 c= Union (b2 (\+\) b3);
:: SETLIM_2:th 43
theorem
for b1 being Element of NAT
for b2 being set
for b3, b4 being Function-like quasi_total Relation of NAT,bool b2 holds
(inferior_setsequence (b3 (/\) b4)) . b1 = ((inferior_setsequence b3) . b1) /\ ((inferior_setsequence b4) . b1);
:: SETLIM_2:th 44
theorem
for b1 being Element of NAT
for b2 being set
for b3, b4 being Function-like quasi_total Relation of NAT,bool b2 holds
((inferior_setsequence b3) . b1) \/ ((inferior_setsequence b4) . b1) c= (inferior_setsequence (b3 (\/) b4)) . b1;
:: SETLIM_2:th 45
theorem
for b1 being Element of NAT
for b2 being set
for b3, b4 being Function-like quasi_total Relation of NAT,bool b2 holds
(inferior_setsequence (b3 (\) b4)) . b1 c= ((inferior_setsequence b3) . b1) \ ((inferior_setsequence b4) . b1);
:: SETLIM_2:th 46
theorem
for b1 being Element of NAT
for b2 being set
for b3, b4 being Function-like quasi_total Relation of NAT,bool b2 holds
(superior_setsequence (b3 (/\) b4)) . b1 c= ((superior_setsequence b3) . b1) /\ ((superior_setsequence b4) . b1);
:: SETLIM_2:th 47
theorem
for b1 being Element of NAT
for b2 being set
for b3, b4 being Function-like quasi_total Relation of NAT,bool b2 holds
(superior_setsequence (b3 (\/) b4)) . b1 = ((superior_setsequence b3) . b1) \/ ((superior_setsequence b4) . b1);
:: SETLIM_2:th 48
theorem
for b1 being Element of NAT
for b2 being set
for b3, b4 being Function-like quasi_total Relation of NAT,bool b2 holds
((superior_setsequence b3) . b1) \ ((superior_setsequence b4) . b1) c= (superior_setsequence (b3 (\) b4)) . b1;
:: SETLIM_2:th 49
theorem
for b1 being Element of NAT
for b2 being set
for b3, b4 being Function-like quasi_total Relation of NAT,bool b2 holds
((superior_setsequence b3) . b1) \+\ ((superior_setsequence b4) . b1) c= (superior_setsequence (b3 (\+\) b4)) . b1;
:: SETLIM_2:th 50
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Element of bool b2
for b4 being Function-like quasi_total Relation of NAT,bool b2 holds
(inferior_setsequence (b3 (/\) b4)) . b1 = b3 /\ ((inferior_setsequence b4) . b1);
:: SETLIM_2:th 51
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Element of bool b2
for b4 being Function-like quasi_total Relation of NAT,bool b2 holds
(inferior_setsequence (b3 (\/) b4)) . b1 = b3 \/ ((inferior_setsequence b4) . b1);
:: SETLIM_2:th 52
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Element of bool b2
for b4 being Function-like quasi_total Relation of NAT,bool b2 holds
(inferior_setsequence (b3 (\) b4)) . b1 c= b3 \ ((inferior_setsequence b4) . b1);
:: SETLIM_2:th 53
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Element of bool b2
for b4 being Function-like quasi_total Relation of NAT,bool b2 holds
(inferior_setsequence (b4 (\) b3)) . b1 = ((inferior_setsequence b4) . b1) \ b3;
:: SETLIM_2:th 54
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Element of bool b2
for b4 being Function-like quasi_total Relation of NAT,bool b2 holds
(inferior_setsequence (b3 (\+\) b4)) . b1 c= b3 \+\ ((inferior_setsequence b4) . b1);
:: SETLIM_2:th 55
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Element of bool b2
for b4 being Function-like quasi_total Relation of NAT,bool b2 holds
(superior_setsequence (b3 (/\) b4)) . b1 = b3 /\ ((superior_setsequence b4) . b1);
:: SETLIM_2:th 56
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Element of bool b2
for b4 being Function-like quasi_total Relation of NAT,bool b2 holds
(superior_setsequence (b3 (\/) b4)) . b1 = b3 \/ ((superior_setsequence b4) . b1);
:: SETLIM_2:th 57
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Element of bool b2
for b4 being Function-like quasi_total Relation of NAT,bool b2 holds
b3 \ ((superior_setsequence b4) . b1) c= (superior_setsequence (b3 (\) b4)) . b1;
:: SETLIM_2:th 58
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Element of bool b2
for b4 being Function-like quasi_total Relation of NAT,bool b2 holds
(superior_setsequence (b4 (\) b3)) . b1 = ((superior_setsequence b4) . b1) \ b3;
:: SETLIM_2:th 59
theorem
for b1 being Element of NAT
for b2 being set
for b3 being Element of bool b2
for b4 being Function-like quasi_total Relation of NAT,bool b2 holds
b3 \+\ ((superior_setsequence b4) . b1) c= (superior_setsequence (b3 (\+\) b4)) . b1;
:: SETLIM_2:th 60
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1 holds
lim_inf (b2 (/\) b3) = (lim_inf b2) /\ lim_inf b3;
:: SETLIM_2:th 61
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1 holds
(lim_inf b2) \/ lim_inf b3 c= lim_inf (b2 (\/) b3);
:: SETLIM_2:th 62
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1 holds
lim_inf (b2 (\) b3) c= (lim_inf b2) \ lim_inf b3;
:: SETLIM_2:th 63
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1
st (b2 is convergent(b1) or b3 is convergent(b1))
holds lim_inf (b2 (\/) b3) = (lim_inf b2) \/ lim_inf b3;
:: SETLIM_2:th 64
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1
st b2 is convergent(b1)
holds lim_inf (b3 (\) b2) = (lim_inf b3) \ lim_inf b2;
:: SETLIM_2:th 65
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1
st (b2 is convergent(b1) or b3 is convergent(b1))
holds lim_inf (b2 (\+\) b3) c= (lim_inf b2) \+\ lim_inf b3;
:: SETLIM_2:th 66
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1
st b2 is convergent(b1) & b3 is convergent(b1)
holds lim_inf (b2 (\+\) b3) = (lim_inf b2) \+\ lim_inf b3;
:: SETLIM_2:th 67
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1 holds
lim_sup (b2 (/\) b3) c= (lim_sup b2) /\ lim_sup b3;
:: SETLIM_2:th 68
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1 holds
lim_sup (b2 (\/) b3) = (lim_sup b2) \/ lim_sup b3;
:: SETLIM_2:th 69
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1 holds
(lim_sup b2) \ lim_sup b3 c= lim_sup (b2 (\) b3);
:: SETLIM_2:th 70
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1 holds
(lim_sup b2) \+\ lim_sup b3 c= lim_sup (b2 (\+\) b3);
:: SETLIM_2:th 71
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1
st (b2 is convergent(b1) or b3 is convergent(b1))
holds lim_sup (b2 (/\) b3) = (lim_sup b2) /\ lim_sup b3;
:: SETLIM_2:th 72
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1
st b2 is convergent(b1)
holds lim_sup (b3 (\) b2) = (lim_sup b3) \ lim_sup b2;
:: SETLIM_2:th 73
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1
st b2 is convergent(b1) & b3 is convergent(b1)
holds lim_sup (b2 (\+\) b3) = (lim_sup b2) \+\ lim_sup b3;
:: SETLIM_2:th 74
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1 holds
lim_inf (b2 (/\) b3) = b2 /\ lim_inf b3;
:: SETLIM_2:th 75
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1 holds
lim_inf (b2 (\/) b3) = b2 \/ lim_inf b3;
:: SETLIM_2:th 76
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1 holds
lim_inf (b2 (\) b3) c= b2 \ lim_inf b3;
:: SETLIM_2:th 77
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1 holds
lim_inf (b3 (\) b2) = (lim_inf b3) \ b2;
:: SETLIM_2:th 78
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1 holds
lim_inf (b2 (\+\) b3) c= b2 \+\ lim_inf b3;
:: SETLIM_2:th 79
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 convergent(b1)
holds lim_inf (b2 (\) b3) = b2 \ lim_inf b3;
:: SETLIM_2:th 80
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 convergent(b1)
holds lim_inf (b2 (\+\) b3) = b2 \+\ lim_inf b3;
:: SETLIM_2:th 81
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1 holds
lim_sup (b2 (/\) b3) = b2 /\ lim_sup b3;
:: SETLIM_2:th 82
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1 holds
lim_sup (b2 (\/) b3) = b2 \/ lim_sup b3;
:: SETLIM_2:th 83
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1 holds
b2 \ lim_sup b3 c= lim_sup (b2 (\) b3);
:: SETLIM_2:th 84
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1 holds
lim_sup (b3 (\) b2) = (lim_sup b3) \ b2;
:: SETLIM_2:th 85
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Function-like quasi_total Relation of NAT,bool b1 holds
b2 \+\ lim_sup b3 c= lim_sup (b2 (\+\) b3);
:: SETLIM_2:th 86
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 convergent(b1)
holds lim_sup (b2 (\) b3) = b2 \ lim_sup b3;
:: SETLIM_2:th 87
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 convergent(b1)
holds lim_sup (b2 (\+\) b3) = b2 \+\ lim_sup b3;
:: SETLIM_2:th 88
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1
st b2 is convergent(b1) & b3 is convergent(b1)
holds b2 (/\) b3 is convergent(b1) &
lim (b2 (/\) b3) = (lim b2) /\ lim b3;
:: SETLIM_2:th 89
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1
st b2 is convergent(b1) & b3 is convergent(b1)
holds b2 (\/) b3 is convergent(b1) &
lim (b2 (\/) b3) = (lim b2) \/ lim b3;
:: SETLIM_2:th 90
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1
st b2 is convergent(b1) & b3 is convergent(b1)
holds b2 (\) b3 is convergent(b1) &
lim (b2 (\) b3) = (lim b2) \ lim b3;
:: SETLIM_2:th 91
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1
st b2 is convergent(b1) & b3 is convergent(b1)
holds b2 (\+\) b3 is convergent(b1) &
lim (b2 (\+\) b3) = (lim b2) \+\ lim b3;
:: SETLIM_2:th 92
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 convergent(b1)
holds b2 (/\) b3 is convergent(b1) & lim (b2 (/\) b3) = b2 /\ lim b3;
:: SETLIM_2:th 93
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 convergent(b1)
holds b2 (\/) b3 is convergent(b1) & lim (b2 (\/) b3) = b2 \/ lim b3;
:: SETLIM_2:th 94
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 convergent(b1)
holds b2 (\) b3 is convergent(b1) & lim (b2 (\) b3) = b2 \ lim b3;
:: SETLIM_2:th 95
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 convergent(b1)
holds b3 (\) b2 is convergent(b1) & lim (b3 (\) b2) = (lim b3) \ b2;
:: SETLIM_2:th 96
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 convergent(b1)
holds b2 (\+\) b3 is convergent(b1) & lim (b2 (\+\) b3) = b2 \+\ lim b3;