Article FRECHET2, MML version 4.99.1005
:: FRECHET2:funcnot 1 => FRECHET2:func 1
definition
let a1 be non empty 1-sorted;
let a2 be Function-like quasi_total Relation of NAT,NAT;
let a3 be Function-like quasi_total Relation of NAT,the carrier of a1;
redefine func a3 * a2 -> Function-like quasi_total Relation of NAT,the carrier of a1;
end;
:: FRECHET2:th 1
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL holds
b2 * b3 is Function-like quasi_total Relation of NAT,the carrier of b1;
:: FRECHET2:th 2
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 = id NAT
holds b1 is Function-like quasi_total natural-valued increasing Relation of NAT,REAL;
:: FRECHET2:th 3
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 is subsequence of b2;
:: FRECHET2:th 4
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being subsequence of b2 holds
rng b3 c= rng b2;
:: FRECHET2:funcnot 2 => FRECHET2:func 2
definition
let a1 be non empty 1-sorted;
let a2 be Function-like quasi_total natural-valued increasing Relation of NAT,REAL;
let a3 be Function-like quasi_total Relation of NAT,the carrier of a1;
redefine func a3 * a2 -> subsequence of a3;
end;
:: FRECHET2:th 5
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being subsequence of b2
for b4 being subsequence of b3 holds
b4 is subsequence of b2;
:: FRECHET2:sch 1
scheme FRECHET2:sch 1
{F1 -> non empty 1-sorted,
F2 -> Function-like quasi_total Relation of NAT,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
ex b3 being Element of the carrier of F1() st
b1 <= b2 & b3 = F2() . b2 & P1[b3];
:: FRECHET2:sch 2
scheme FRECHET2:sch 2
{F1 -> non empty TopStruct,
F2 -> Function-like quasi_total Relation of NAT,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
ex b3 being Element of the carrier of F1() st
b1 <= b2 & b3 = F2() . b2 & P1[b3];
:: FRECHET2:th 6
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of bool the carrier of b1
st for b4 being subsequence of b2 holds
not rng b4 c= b3
holds ex b4 being Element of NAT st
for b5 being Element of NAT
st b4 <= b5
holds not b2 . b5 in b3;
:: FRECHET2:th 7
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3, b4 being Element of bool the carrier of b1
st rng b2 c= b3 \/ b4
holds ex b5 being subsequence of b2 st
(rng b5 c= b3 or rng b5 c= b4);
:: FRECHET2:th 8
theorem
for b1 being non empty TopSpace-like TopStruct
st for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3, b4 being Element of the carrier of b1
st b3 in Lim b2 & b4 in Lim b2
holds b3 = b4
holds b1 is being_T1;
:: FRECHET2:th 9
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is being_T2
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3, b4 being Element of the carrier of b1
st b3 in Lim b2 & b4 in Lim b2
holds b3 = b4;
:: FRECHET2:th 10
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is first-countable
holds b1 is being_T2
iff
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3, b4 being Element of the carrier of b1
st b3 in Lim b2 & b4 in Lim b2
holds b3 = b4;
:: FRECHET2:th 11
theorem
for b1 being non empty TopStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is not convergent(b1)
holds Lim b2 = {};
:: FRECHET2:th 12
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is closed(b1)
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b3 c= b2
holds Lim b3 c= b2;
:: FRECHET2:th 13
theorem
for b1 being non empty TopStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of the carrier of b1
st not b2 is_convergent_to b3
holds ex b4 being subsequence of b2 st
for b5 being subsequence of b4 holds
not b5 is_convergent_to b3;
:: FRECHET2:th 14
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b3 is continuous(b1, b2)
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
for b5 being Function-like quasi_total Relation of NAT,the carrier of b2
st b5 = b3 * b4
holds b3 .: Lim b4 c= Lim b5;
:: FRECHET2:th 15
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b1 is sequential
holds b3 is continuous(b1, b2)
iff
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
for b5 being Function-like quasi_total Relation of NAT,the carrier of b2
st b5 = b3 * b4
holds b3 .: Lim b4 c= Lim b5;
:: FRECHET2:funcnot 3 => FRECHET2:func 3
definition
let a1 be non empty TopStruct;
let a2 be Element of bool the carrier of a1;
func Cl_Seq A2 -> Element of bool the carrier of a1 means
for b1 being Element of the carrier of a1 holds
b1 in it
iff
ex b2 being Function-like quasi_total Relation of NAT,the carrier of a1 st
rng b2 c= a2 & b1 in Lim b2;
end;
:: FRECHET2:def 2
theorem
for b1 being non empty TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
b3 = Cl_Seq b2
iff
for b4 being Element of the carrier of b1 holds
b4 in b3
iff
ex b5 being Function-like quasi_total Relation of NAT,the carrier of b1 st
rng b5 c= b2 & b4 in Lim b5;
:: FRECHET2:th 16
theorem
for b1 being non empty TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
for b4 being Element of the carrier of b1
st rng b3 c= b2 & b4 in Lim b3
holds b4 in Cl b2;
:: FRECHET2:th 17
theorem
for b1 being non empty TopStruct
for b2 being Element of bool the carrier of b1 holds
Cl_Seq b2 c= Cl b2;
:: FRECHET2:th 18
theorem
for b1 being non empty TopStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being subsequence of b2
for b4 being Element of the carrier of b1
st b2 is_convergent_to b4
holds b3 is_convergent_to b4;
:: FRECHET2:th 19
theorem
for b1 being non empty TopStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being subsequence of b2 holds
Lim b2 c= Lim b3;
:: FRECHET2:th 20
theorem
for b1 being non empty TopStruct holds
Cl_Seq {} b1 = {};
:: FRECHET2:th 21
theorem
for b1 being non empty TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 c= Cl_Seq b2;
:: FRECHET2:th 22
theorem
for b1 being non empty TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
(Cl_Seq b2) \/ Cl_Seq b3 = Cl_Seq (b2 \/ b3);
:: FRECHET2:th 23
theorem
for b1 being non empty TopStruct holds
b1 is Frechet
iff
for b2 being Element of bool the carrier of b1 holds
Cl b2 = Cl_Seq b2;
:: FRECHET2:th 24
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is Frechet
for b2, b3 being Element of bool the carrier of b1 holds
Cl_Seq {} b1 = {} &
b2 c= Cl_Seq b2 &
Cl_Seq (b2 \/ b3) = (Cl_Seq b2) \/ Cl_Seq b3 &
Cl_Seq Cl_Seq b2 = Cl_Seq b2;
:: FRECHET2:th 25
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is sequential &
(for b2 being Element of bool the carrier of b1 holds
Cl_Seq Cl_Seq b2 = Cl_Seq b2)
holds b1 is Frechet;
:: FRECHET2:th 26
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is sequential
holds b1 is Frechet
iff
for b2, b3 being Element of bool the carrier of b1 holds
Cl_Seq {} b1 = {} &
b2 c= Cl_Seq b2 &
Cl_Seq (b2 \/ b3) = (Cl_Seq b2) \/ Cl_Seq b3 &
Cl_Seq Cl_Seq b2 = Cl_Seq b2;
:: FRECHET2:funcnot 4 => FRECHET2:func 4
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
assume ex b1 being Element of the carrier of a1 st
Lim a2 = {b1};
func lim A2 -> Element of the carrier of a1 means
a2 is_convergent_to it;
end;
:: FRECHET2:def 3
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st ex b3 being Element of the carrier of b1 st
Lim b2 = {b3}
for b3 being Element of the carrier of b1 holds
b3 = lim b2
iff
b2 is_convergent_to b3;
:: FRECHET2:th 27
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is being_T2
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
st b2 is convergent(b1)
holds ex b3 being Element of the carrier of b1 st
Lim b2 = {b3};
:: FRECHET2:th 28
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is being_T2
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of the carrier of b1 holds
b2 is_convergent_to b3
iff
b2 is convergent(b1) & b3 = lim b2;
:: FRECHET2:th 29
theorem
for b1 being MetrStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 is Function-like quasi_total Relation of NAT,the carrier of TopSpaceMetr b1;
:: FRECHET2:th 30
theorem
for b1 being non empty MetrStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of TopSpaceMetr b1 holds
b2 is Function-like quasi_total Relation of NAT,the carrier of b1;
:: FRECHET2:th 31
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Function-like quasi_total Relation of NAT,the carrier of TopSpaceMetr b1
for b5 being Element of the carrier of TopSpaceMetr b1
st b2 = b4 & b3 = b5
holds b2 is_convergent_in_metrspace_to b3
iff
b4 is_convergent_to b5;
:: FRECHET2:th 32
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of TopSpaceMetr b1
st b2 = b3
holds b2 is convergent(b1)
iff
b3 is convergent(TopSpaceMetr b1);
:: FRECHET2:th 33
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of TopSpaceMetr b1
st b2 = b3 & b2 is convergent(b1)
holds lim b2 = lim b3;
:: FRECHET2:prednot 1 => FRECHET2:pred 1
definition
let a1 be TopStruct;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
let a3 be Element of the carrier of a1;
pred A3 is_a_cluster_point_of A2 means
for b1 being Element of bool the carrier of a1
for b2 being Element of NAT
st b1 is open(a1) & a3 in b1
holds ex b3 being Element of NAT st
b2 <= b3 & a2 . b3 in b1;
end;
:: FRECHET2:dfs 3
definiens
let a1 be TopStruct;
let a2 be Function-like quasi_total Relation of NAT,the carrier of 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 Element of bool the carrier of a1
for b2 being Element of NAT
st b1 is open(a1) & a3 in b1
holds ex b3 being Element of NAT st
b2 <= b3 & a2 . b3 in b1;
:: FRECHET2:def 4
theorem
for b1 being TopStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of the carrier of b1 holds
b3 is_a_cluster_point_of b2
iff
for b4 being Element of bool the carrier of b1
for b5 being Element of NAT
st b4 is open(b1) & b3 in b4
holds ex b6 being Element of NAT st
b5 <= b6 & b2 . b6 in b4;
:: FRECHET2:th 34
theorem
for b1 being non empty TopStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of the carrier of b1
st ex b4 being subsequence of b2 st
b4 is_convergent_to b3
holds b3 is_a_cluster_point_of b2;
:: FRECHET2:th 35
theorem
for b1 being non empty TopStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of the carrier of b1
st b2 is_convergent_to b3
holds b3 is_a_cluster_point_of b2;
:: FRECHET2:th 36
theorem
for b1 being non empty TopStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1
st b4 = {b5 where b5 is Element of the carrier of b1: b3 in Cl {b5}} &
rng b2 c= b4
holds b2 is_convergent_to b3;
:: FRECHET2:th 37
theorem
for b1 being non empty TopStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3, b4 being Element of the carrier of b1
st for b5 being Element of NAT holds
b2 . b5 = b4 & b2 is_convergent_to b3
holds b3 in Cl {b4};
:: FRECHET2:th 38
theorem
for b1 being non empty TopStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
st b3 = {b5 where b5 is Element of the carrier of b1: b2 in Cl {b5}} &
rng b4 misses b3 &
b4 is_convergent_to b2
holds ex b5 being subsequence of b4 st
b5 is one-to-one;
:: FRECHET2:th 39
theorem
for b1 being non empty TopStruct
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b3 c= rng b2 & b3 is one-to-one
holds ex b4 being Function-like quasi_total bijective Relation of NAT,NAT st
b3 * b4 is subsequence of b2;
:: FRECHET2:sch 3
scheme FRECHET2:sch 3
{F1 -> non empty 1-sorted,
F2 -> Function-like quasi_total Relation of NAT,the carrier of F1(),
F3 -> Function-like quasi_total bijective Relation of NAT,NAT}:
ex b1 being Element of NAT st
for b2 being Element of NAT
st b1 <= b2
holds P1[(F2() * F3()) . b2]
provided
ex b1 being Element of NAT st
for b2 being Element of NAT
for b3 being Element of the carrier of F1()
st b1 <= b2 & b3 = F2() . b2
holds P1[b3];
:: FRECHET2:sch 4
scheme FRECHET2:sch 4
{F1 -> non empty TopStruct,
F2 -> Function-like quasi_total Relation of NAT,the carrier of F1(),
F3 -> Function-like quasi_total bijective Relation of NAT,NAT}:
ex b1 being Element of NAT st
for b2 being Element of NAT
st b1 <= b2
holds P1[(F2() * F3()) . b2]
provided
ex b1 being Element of NAT st
for b2 being Element of NAT
for b3 being Element of the carrier of F1()
st b1 <= b2 & b3 = F2() . b2
holds P1[b3];
:: FRECHET2:th 40
theorem
for b1 being non empty TopStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total bijective Relation of NAT,NAT
for b4 being Element of the carrier of b1
st b2 is_convergent_to b4
holds b2 * b3 is_convergent_to b4;
:: FRECHET2:th 41
theorem
for b1 being Element of NAT holds
ex b2 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL st
for b3 being Element of NAT holds
b2 . b3 = b3 + b1;
:: FRECHET2:th 42
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of NAT holds
ex b4 being subsequence of b2 st
for b5 being Element of NAT holds
b4 . b5 = b2 . (b5 + b3);
:: FRECHET2:th 43
theorem
for b1 being non empty TopStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of the carrier of b1
for b4 being subsequence of b2
st b3 is_a_cluster_point_of b2 &
(ex b5 being Element of NAT st
for b6 being Element of NAT holds
b4 . b6 = b2 . (b6 + b5))
holds b3 is_a_cluster_point_of b4;
:: FRECHET2:th 44
theorem
for b1 being non empty TopStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of the carrier of b1
st b3 is_a_cluster_point_of b2
holds b3 in Cl rng b2;
:: FRECHET2:th 45
theorem
for b1 being non empty TopStruct
st b1 is Frechet
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of the carrier of b1
st b3 is_a_cluster_point_of b2
holds ex b4 being subsequence of b2 st
b4 is_convergent_to b3;
:: FRECHET2:th 46
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is first-countable
for b2 being Element of the carrier of b1 holds
ex b3 being Basis of b2 st
ex b4 being Relation-like Function-like set st
proj1 b4 = NAT &
proj2 b4 = b3 &
(for b5, b6 being Element of NAT
st b5 <= b6
holds b4 . b6 c= b4 . b5);
:: FRECHET2:th 47
theorem
for b1 being non empty TopSpace-like TopStruct
st for b2 being Element of the carrier of b1 holds
Cl {b2} = {b2}
holds b1 is being_T1;
:: FRECHET2:th 48
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is being_T2
holds b1 is being_T1;
:: FRECHET2:th 49
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is not being_T1
holds ex b2, b3 being Element of the carrier of b1 st
ex b4 being Function-like quasi_total Relation of NAT,the carrier of b1 st
b4 = NAT --> b2 & b2 <> b3 & b4 is_convergent_to b3;