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;