Article KURATO_2, MML version 4.99.1005
:: KURATO_2:th 2
theorem
for b1 being Relation-like Function-like set
for b2 being set
st b2 in proj1 b1
holds meet b1 c= b1 . b2;
:: KURATO_2:th 3
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1 holds
b2 = b3
iff
for b4 being Element of NAT holds
b2 . b4 = b3 . b4;
:: KURATO_2:th 4
theorem
for b1, b2, b3, b4 being set
st b1 meets b2 & b3 meets b4
holds [:b1,b3:] meets [:b2,b4:];
:: KURATO_2:condreg 1
registration
let a1 be set;
cluster Function-like quasi_total -> non empty (Relation of NAT,bool a1);
end;
:: KURATO_2:exreg 1
registration
let a1 be non empty set;
cluster non empty Relation-like non-empty Function-like quasi_total total Relation of NAT,bool a1;
end;
:: KURATO_2:modenot 1
definition
let a1 be 1-sorted;
mode SetSequence of a1 is Function-like quasi_total Relation of NAT,bool the carrier of a1;
end;
:: KURATO_2:sch 1
scheme KURATO_2:sch 1
{F1 -> set,
F2 -> Element of bool F1()}:
ex b1 being Function-like quasi_total Relation of NAT,bool F1() st
for b2 being Element of NAT holds
b1 . b2 = F2(b2)
:: KURATO_2:funcnot 1 => KURATO_2:func 1
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
redefine func Union a2 -> Element of bool a1;
end;
:: KURATO_2:funcnot 2 => KURATO_2:func 2
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
redefine func meet a2 -> Element of bool a1;
end;
:: KURATO_2:funcnot 3 => KURATO_2:func 3
definition
let a1 be non empty set;
let a2 be Function-like quasi_total Relation of NAT,a1;
let a3 be Element of NAT;
func A2 ^\ A3 -> Function-like quasi_total Relation of NAT,a1 means
for b1 being Element of NAT holds
it . b1 = a2 . (b1 + a3);
end;
:: KURATO_2:def 2
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of NAT,b1
for b3 being Element of NAT
for b4 being Function-like quasi_total Relation of NAT,b1 holds
b4 = b2 ^\ b3
iff
for b5 being Element of NAT holds
b4 . b5 = b2 . (b5 + b3);
:: KURATO_2:funcnot 4 => 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 means
ex b1 being Function-like quasi_total Relation of NAT,bool a1 st
it = Union b1 &
(for b2 being Element of NAT holds
b1 . b2 = meet (a2 ^\ b2));
end;
:: KURATO_2:def 3
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
for b3 being Element of bool b1 holds
b3 = lim_inf b2
iff
ex b4 being Function-like quasi_total Relation of NAT,bool b1 st
b3 = Union b4 &
(for b5 being Element of NAT holds
b4 . b5 = meet (b2 ^\ b5));
:: KURATO_2:funcnot 5 => 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 means
ex b1 being Function-like quasi_total Relation of NAT,bool a1 st
it = meet b1 &
(for b2 being Element of NAT holds
b1 . b2 = Union (a2 ^\ b2));
end;
:: KURATO_2:def 4
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
for b3 being Element of bool b1 holds
b3 = lim_sup b2
iff
ex b4 being Function-like quasi_total Relation of NAT,bool b1 st
b3 = meet b4 &
(for b5 being Element of NAT holds
b4 . b5 = Union (b2 ^\ b5));
:: KURATO_2:th 6
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
for b3 being set holds
b3 in meet b2
iff
for b4 being Element of NAT holds
b3 in b2 . b4;
:: KURATO_2:th 7
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
for b3 being set holds
b3 in lim_inf b2
iff
ex b4 being Element of NAT st
for b5 being Element of NAT holds
b3 in b2 . (b4 + b5);
:: KURATO_2:th 8
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
for b3 being set holds
b3 in lim_sup b2
iff
for b4 being Element of NAT holds
ex b5 being Element of NAT st
b3 in b2 . (b4 + b5);
:: KURATO_2:th 9
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
lim_inf b2 c= lim_sup b2;
:: KURATO_2:th 10
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
meet b2 c= lim_inf b2;
:: KURATO_2:th 11
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
lim_sup b2 c= Union b2;
:: KURATO_2:th 12
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
lim_inf b2 = (lim_sup Complement b2) `;
:: KURATO_2:th 13
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
b4 . b5 = (b2 . b5) /\ (b3 . b5)
holds lim_inf b4 = (lim_inf b2) /\ lim_inf b3;
:: KURATO_2:th 14
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
b4 . b5 = (b2 . b5) \/ (b3 . b5)
holds lim_sup b4 = (lim_sup b2) \/ lim_sup b3;
:: KURATO_2:th 15
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
b4 . b5 = (b2 . b5) \/ (b3 . b5)
holds (lim_inf b2) \/ lim_inf b3 c= lim_inf b4;
:: KURATO_2:th 16
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
b4 . b5 = (b2 . b5) /\ (b3 . b5)
holds lim_sup b4 c= (lim_sup b2) /\ lim_sup b3;
:: KURATO_2:th 17
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
for b3 being Element of bool b1
st for b4 being Element of NAT holds
b2 . b4 = b3
holds lim_sup b2 = b3;
:: KURATO_2:th 18
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
for b3 being Element of bool b1
st for b4 being Element of NAT holds
b2 . b4 = b3
holds lim_inf b2 = b3;
:: KURATO_2:th 19
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1
for b4 being Element of bool b1
st for b5 being Element of NAT holds
b3 . b5 = b4 \+\ (b2 . b5)
holds b4 \+\ lim_inf b2 c= lim_sup b3;
:: KURATO_2:th 20
theorem
for b1 being set
for b2, b3 being Function-like quasi_total Relation of NAT,bool b1
for b4 being Element of bool b1
st for b5 being Element of NAT holds
b3 . b5 = b4 \+\ (b2 . b5)
holds b4 \+\ lim_sup b2 c= lim_sup b3;
:: KURATO_2:attrnot 1 => KURATO_2:attr 1
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
attr a2 is descending means
for b1 being Element of NAT holds
a2 . (b1 + 1) c= a2 . b1;
end;
:: KURATO_2:dfs 4
definiens
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
To prove
a2 is descending
it is sufficient to prove
thus for b1 being Element of NAT holds
a2 . (b1 + 1) c= a2 . b1;
:: KURATO_2:def 5
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
b2 is descending(b1)
iff
for b3 being Element of NAT holds
b2 . (b3 + 1) c= b2 . b3;
:: KURATO_2:attrnot 2 => KURATO_2:attr 2
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
attr a2 is ascending means
for b1 being Element of NAT holds
a2 . b1 c= a2 . (b1 + 1);
end;
:: KURATO_2:dfs 5
definiens
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
To prove
a2 is ascending
it is sufficient to prove
thus for b1 being Element of NAT holds
a2 . b1 c= a2 . (b1 + 1);
:: KURATO_2:def 6
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
b2 is ascending(b1)
iff
for b3 being Element of NAT holds
b2 . b3 c= b2 . (b3 + 1);
:: KURATO_2:th 21
theorem
for b1 being Relation-like Function-like set
st for b2 being Element of NAT holds
b1 . (b2 + 1) c= b1 . b2
for b2, b3 being Element of NAT
st b2 <= b3
holds b1 . b3 c= b1 . b2;
:: KURATO_2:th 22
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
st b2 is descending(b1)
for b3, b4 being Element of NAT
st b4 <= b3
holds b2 . b3 c= b2 . b4;
:: KURATO_2:th 23
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
st b2 is ascending(b1)
for b3, b4 being Element of NAT
st b4 <= b3
holds b2 . b4 c= b2 . b3;
:: KURATO_2:th 24
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
for b3 being set
st b2 is descending(b1) &
(ex b4 being Element of NAT st
for b5 being Element of NAT
st b4 < b5
holds b3 in b2 . b5)
holds b3 in meet b2;
:: KURATO_2:th 25
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
st b2 is descending(b1)
holds lim_inf b2 = meet b2;
:: KURATO_2:th 26
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
st b2 is ascending(b1)
holds lim_sup b2 = Union b2;
:: KURATO_2:attrnot 3 => KURATO_2:attr 3
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
attr a2 is convergent means
lim_sup a2 = lim_inf a2;
end;
:: KURATO_2:dfs 6
definiens
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
To prove
a2 is convergent
it is sufficient to prove
thus lim_sup a2 = lim_inf a2;
:: KURATO_2:def 7
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
b2 is convergent(b1)
iff
lim_sup b2 = lim_inf b2;
:: KURATO_2:th 27
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1
st b2 is constant
holds the_value_of b2 is Element of bool b1;
:: KURATO_2:attrnot 4 => KURATO_2:attr 4
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
redefine attr a2 is constant means
ex b1 being Element of bool a1 st
for b2 being Element of NAT holds
a2 . b2 = b1;
end;
:: KURATO_2:dfs 7
definiens
let a1 be set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
To prove
a1 is constant
it is sufficient to prove
thus ex b1 being Element of bool a1 st
for b2 being Element of NAT holds
a2 . b2 = b1;
:: KURATO_2:def 8
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
b2 is constant
iff
ex b3 being Element of bool b1 st
for b4 being Element of NAT holds
b2 . b4 = b3;
:: KURATO_2:condreg 2
registration
let a1 be set;
cluster Function-like constant quasi_total -> descending ascending convergent (Relation of NAT,bool a1);
end;
:: KURATO_2:exreg 2
registration
let a1 be set;
cluster non empty Relation-like Function-like constant quasi_total total Relation of NAT,bool a1;
end;
:: KURATO_2:funcnot 6 => KURATO_2:func 6
definition
let a1 be set;
let a2 be Function-like quasi_total convergent Relation of NAT,bool a1;
func Lim_K A2 -> Element of bool a1 means
it = lim_sup a2 & it = lim_inf a2;
end;
:: KURATO_2:def 9
theorem
for b1 being set
for b2 being Function-like quasi_total convergent Relation of NAT,bool b1
for b3 being Element of bool b1 holds
b3 = Lim_K b2
iff
b3 = lim_sup b2 & b3 = lim_inf b2;
:: KURATO_2:th 28
theorem
for b1 being set
for b2 being Function-like quasi_total convergent Relation of NAT,bool b1
for b3 being set holds
b3 in Lim_K b2
iff
ex b4 being Element of NAT st
for b5 being Element of NAT holds
b3 in b2 . (b4 + b5);
:: KURATO_2:funcreg 1
registration
let a1 be FinSequence of the carrier of TOP-REAL 2;
cluster L~ a1 -> closed;
end;
:: KURATO_2:th 30
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of Euclid b1
for b3 being real set holds
Ball(b2,b3) is open Element of bool the carrier of TOP-REAL b1;
:: KURATO_2:th 32
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of Euclid b1
for b3, b4 being Element of the carrier of TOP-REAL b1
for b5 being real set
st b2 = b4 & b3 in Ball(b2,b5)
holds |.b3 - b4.| < b5;
:: KURATO_2:th 33
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of Euclid b1
for b3, b4 being Element of the carrier of TOP-REAL b1
for b5 being real set
st b2 = b4 & |.b3 - b4.| < b5
holds b3 in Ball(b2,b5);
:: KURATO_2:th 34
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1
for b3 being Element of bool the carrier of TOP-REAL b1
st b2 in Cl b3
holds ex b4 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 st
rng b4 c= b3 & b4 is convergent(b1) & lim b4 = b2;
:: KURATO_2:funcreg 2
registration
let a1 be non empty Reflexive discerning symmetric triangle MetrStruct;
cluster TopSpaceMetr a1 -> first-countable;
end;
:: KURATO_2:funcreg 3
registration
let a1 be Element of NAT;
cluster TOP-REAL a1 -> strict TopSpace-like first-countable;
end;
:: KURATO_2:th 36
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3 being Element of the carrier of TOP-REAL b1
for b4 being Element of the carrier of Euclid b1
st b3 = b4
holds b3 in Cl b2
iff
for b5 being real set
st 0 < b5
holds Ball(b4,b5) meets b2;
:: KURATO_2:th 37
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4 being Element of the carrier of Euclid b1
st b4 = b2 & b2 <> b3
holds ex b5 being Element of REAL st
not b3 in Ball(b4,b5);
:: KURATO_2:th 38
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1 holds
b2 is not Bounded(b1)
iff
for b3 being Element of REAL
st 0 < b3
holds ex b4, b5 being Element of the carrier of Euclid b1 st
b4 in b2 & b5 in b2 & b3 < dist(b4,b5);
:: KURATO_2:th 39
theorem
for b1 being Element of NAT
for b2, b3 being real set
for b4, b5 being Element of the carrier of Euclid b1
st Ball(b4,b2) meets Ball(b5,b3)
holds dist(b4,b5) < b2 + b3;
:: KURATO_2:th 40
theorem
for b1 being Element of NAT
for b2, b3, b4 being real set
for b5, b6, b7 being Element of the carrier of Euclid b1
st Ball(b5,b2) meets Ball(b7,b4) & Ball(b7,b4) meets Ball(b6,b3)
holds dist(b5,b6) < (b2 + b3) + (2 * b4);
:: KURATO_2:th 41
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2
for b5 being Element of bool the carrier of [:b1,b2:] holds
b5 is a_neighborhood of [:{b3},{b4}:]
iff
b5 is a_neighborhood of [b3,b4];
:: KURATO_2:funcnot 7 => KURATO_2:func 7
definition
let a1 be non empty set;
let a2 be Function-like quasi_total Relation of NAT,bool a1;
let a3 be Element of NAT;
redefine func a2 . a3 -> Element of bool a1;
end;
:: KURATO_2:th 42
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1
for b3 being Function-like quasi_total natural-valued Relation of NAT,REAL holds
b2 * b3 is Function-like quasi_total Relation of NAT,bool the carrier of b1;
:: KURATO_2:th 43
theorem
id NAT is Function-like quasi_total natural-valued increasing Relation of NAT,REAL;
:: KURATO_2:modenot 2 => KURATO_2:mode 1
definition
let a1 be non empty 1-sorted;
let a2 be Function-like quasi_total Relation of NAT,bool the carrier of a1;
mode subsequence of A2 -> Function-like quasi_total Relation of NAT,bool the carrier of a1 means
ex b1 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL st
it = a2 * b1;
end;
:: KURATO_2:dfs 9
definiens
let a1 be non empty 1-sorted;
let a2, a3 be Function-like quasi_total Relation of NAT,bool the carrier of a1;
To prove
a3 is subsequence of a2
it is sufficient to prove
thus ex b1 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL st
a3 = a2 * b1;
:: KURATO_2:def 10
theorem
for b1 being non empty 1-sorted
for b2, b3 being Function-like quasi_total Relation of NAT,bool the carrier of b1 holds
b3 is subsequence of b2
iff
ex b4 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL st
b3 = b2 * b4;
:: KURATO_2:th 44
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1 holds
b2 is subsequence of b2;
:: KURATO_2:th 45
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1
for b3 being subsequence of b2 holds
rng b3 c= rng b2;
:: KURATO_2:th 46
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1
for b3 being subsequence of b2
for b4 being subsequence of b3 holds
b4 is subsequence of b2;
:: KURATO_2:th 47
theorem
for b1 being non empty 1-sorted
for b2, b3 being Function-like quasi_total Relation of NAT,bool the carrier of b1
for b4 being Element of bool the carrier of b1
st b3 is subsequence of b2 &
(for b5 being Element of NAT holds
b2 . b5 = b4)
holds b3 = b2;
:: KURATO_2:th 48
theorem
for b1 being non empty 1-sorted
for b2 being Function-like constant quasi_total Relation of NAT,bool the carrier of b1
for b3 being subsequence of b2 holds
b2 = b3;
:: KURATO_2:th 49
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1
for b3 being subsequence of b2
for b4 being Element of NAT holds
ex b5 being Element of NAT st
b4 <= b5 & b3 . b4 = b2 . b5;
:: KURATO_2:condreg 3
registration
let a1 be non empty 1-sorted;
let a2 be Function-like constant quasi_total Relation of NAT,bool the carrier of a1;
cluster -> constant (subsequence of a2);
end;
:: KURATO_2:sch 2
scheme KURATO_2:sch 2
{F1 -> non empty TopSpace-like TopStruct,
F2 -> Function-like quasi_total Relation of NAT,bool the carrier of F1()}:
ex b1 being subsequence of F2() st
for b2 being Element of NAT holds
P1[b1 . b2]
provided
for b1 being Element of NAT holds
ex b2 being Element of NAT st
b1 <= b2 & P1[F2() . b2];
:: KURATO_2:funcnot 8 => KURATO_2:func 8
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be Function-like quasi_total Relation of NAT,bool the carrier of a1;
func Lim_inf A2 -> Element of bool the carrier of a1 means
for b1 being Element of the carrier of a1 holds
b1 in it
iff
for b2 being a_neighborhood of b1 holds
ex b3 being Element of NAT st
for b4 being Element of NAT
st b3 < b4
holds a2 . b4 meets b2;
end;
:: KURATO_2:def 11
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1
for b3 being Element of bool the carrier of b1 holds
b3 = Lim_inf b2
iff
for b4 being Element of the carrier of b1 holds
b4 in b3
iff
for b5 being a_neighborhood of b4 holds
ex b6 being Element of NAT st
for b7 being Element of NAT
st b6 < b7
holds b2 . b7 meets b5;
:: KURATO_2:th 50
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL b1
for b3 being Element of the carrier of TOP-REAL b1
for b4 being Element of the carrier of Euclid b1
st b3 = b4
holds b3 in Lim_inf b2
iff
for b5 being real set
st 0 < b5
holds ex b6 being Element of NAT st
for b7 being Element of NAT
st b6 < b7
holds b2 . b7 meets Ball(b4,b5);
:: KURATO_2:th 51
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1 holds
Cl Lim_inf b2 = Lim_inf b2;
:: KURATO_2:th 52
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1 holds
Lim_inf b2 is closed(b1);
:: KURATO_2:th 53
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Function-like quasi_total Relation of NAT,bool the carrier of b1
st b2 is subsequence of b3
holds Lim_inf b3 c= Lim_inf b2;
:: KURATO_2:th 54
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Function-like quasi_total Relation of NAT,bool the carrier of b1
st for b4 being Element of NAT holds
b2 . b4 c= b3 . b4
holds Lim_inf b2 c= Lim_inf b3;
:: KURATO_2:th 55
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Function-like quasi_total Relation of NAT,bool the carrier of b1
st for b5 being Element of NAT holds
b4 . b5 = (b2 . b5) \/ (b3 . b5)
holds (Lim_inf b2) \/ Lim_inf b3 c= Lim_inf b4;
:: KURATO_2:th 56
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Function-like quasi_total Relation of NAT,bool the carrier of b1
st for b5 being Element of NAT holds
b4 . b5 = (b2 . b5) /\ (b3 . b5)
holds Lim_inf b4 c= (Lim_inf b2) /\ Lim_inf b3;
:: KURATO_2:th 57
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Function-like quasi_total Relation of NAT,bool the carrier of b1
st for b4 being Element of NAT holds
b3 . b4 = Cl (b2 . b4)
holds Lim_inf b3 = Lim_inf b2;
:: KURATO_2:th 58
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL b1
for b3 being Element of the carrier of TOP-REAL b1
st ex b4 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1 st
b4 is convergent(b1) &
(for b5 being Element of NAT holds
b4 . b5 in b2 . b5) &
b3 = lim b4
holds b3 in Lim_inf b2;
:: KURATO_2:th 59
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,bool the carrier of b1
st for b4 being Element of NAT holds
b3 . b4 c= b2
holds Lim_inf b3 c= Cl b2;
:: KURATO_2:th 60
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1
for b3 being Element of bool the carrier of b1
st for b4 being Element of NAT holds
b2 . b4 = b3
holds Lim_inf b2 = Cl b3;
:: KURATO_2:th 61
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1
for b3 being closed Element of bool the carrier of b1
st for b4 being Element of NAT holds
b2 . b4 = b3
holds Lim_inf b2 = b3;
:: KURATO_2:th 62
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL b1
for b3 being Element of bool the carrier of TOP-REAL b1
st b3 is Bounded(b1) &
(for b4 being Element of NAT holds
b2 . b4 c= b3)
holds Lim_inf b2 is Bounded(b1);
:: KURATO_2:th 63
theorem
for b1 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
st b2 is Bounded(2) &
(for b3 being Element of NAT holds
b1 . b3 c= b2)
holds Lim_inf b1 is compact(TOP-REAL 2);
:: KURATO_2:th 64
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL b1
for b4 being Function-like quasi_total Relation of NAT,bool the carrier of [:TOP-REAL b1,TOP-REAL b1:]
st for b5 being Element of NAT holds
b4 . b5 = [:b2 . b5,b3 . b5:]
holds [:Lim_inf b2,Lim_inf b3:] = Lim_inf b4;
:: KURATO_2:th 65
theorem
for b1 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL 2 holds
lim_inf b1 c= Lim_inf b1;
:: KURATO_2:th 66
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
Fr ((UBD L~ Cage(b1,b2)) `) = L~ Cage(b1,b2);
:: KURATO_2:funcnot 9 => KURATO_2:func 9
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be Function-like quasi_total Relation of NAT,bool the carrier of a1;
func Lim_sup A2 -> Element of bool the carrier of a1 means
for b1 being set holds
b1 in it
iff
ex b2 being subsequence of a2 st
b1 in Lim_inf b2;
end;
:: KURATO_2:def 12
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1
for b3 being Element of bool the carrier of b1 holds
b3 = Lim_sup b2
iff
for b4 being set holds
b4 in b3
iff
ex b5 being subsequence of b2 st
b4 in Lim_inf b5;
:: KURATO_2:th 67
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of TOP-REAL b1
for b3 being Element of the carrier of TOP-REAL b1
for b4 being Element of the carrier of Euclid b1
st b3 = b4
holds b3 is_a_cluster_point_of b2
iff
for b5 being real set
for b6 being Element of NAT
st 0 < b5
holds ex b7 being Element of NAT st
b6 <= b7 & b2 . b7 in Ball(b4,b5);
:: KURATO_2:th 68
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1 holds
Lim_inf b2 c= Lim_sup b2;
:: KURATO_2:th 69
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL 2
st (for b4 being Element of NAT holds
b1 . b4 c= b2 . b4) &
b3 is subsequence of b1
holds ex b4 being subsequence of b2 st
for b5 being Element of NAT holds
b3 . b5 c= b4 . b5;
:: KURATO_2:th 70
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL 2
st (for b4 being Element of NAT holds
b1 . b4 c= b2 . b4) &
b3 is subsequence of b2
holds ex b4 being subsequence of b1 st
for b5 being Element of NAT holds
b4 . b5 c= b3 . b5;
:: KURATO_2:th 71
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL 2
st for b3 being Element of NAT holds
b1 . b3 c= b2 . b3
holds Lim_sup b1 c= Lim_sup b2;
:: KURATO_2:th 72
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL 2
st for b4 being Element of NAT holds
b3 . b4 = (b1 . b4) \/ (b2 . b4)
holds (Lim_sup b1) \/ Lim_sup b2 c= Lim_sup b3;
:: KURATO_2:th 73
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL 2
st for b4 being Element of NAT holds
b3 . b4 = (b1 . b4) /\ (b2 . b4)
holds Lim_sup b3 c= (Lim_sup b1) /\ Lim_sup b2;
:: KURATO_2:th 74
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL 2
for b3, b4 being Function-like quasi_total Relation of NAT,bool the carrier of [:TOP-REAL 2,TOP-REAL 2:]
st (for b5 being Element of NAT holds
b3 . b5 = [:b1 . b5,b2 . b5:]) &
b4 is subsequence of b3
holds ex b5, b6 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL 2 st
b5 is subsequence of b1 &
b6 is subsequence of b2 &
(for b7 being Element of NAT holds
b4 . b7 = [:b5 . b7,b6 . b7:]);
:: KURATO_2:th 75
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of NAT,bool the carrier of [:TOP-REAL 2,TOP-REAL 2:]
st for b4 being Element of NAT holds
b3 . b4 = [:b1 . b4,b2 . b4:]
holds Lim_sup b3 c= [:Lim_sup b1,Lim_sup b2:];
:: KURATO_2:th 76
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1
for b3 being Element of bool the carrier of b1
st for b4 being Element of NAT holds
b2 . b4 = b3
holds Lim_inf b2 = Lim_sup b2;
:: KURATO_2:th 77
theorem
for b1 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
st for b3 being Element of NAT holds
b1 . b3 = b2
holds Lim_sup b1 = Cl b2;
:: KURATO_2:th 78
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL 2
st for b3 being Element of NAT holds
b2 . b3 = Cl (b1 . b3)
holds Lim_sup b2 = Lim_sup b1;