Article FRECHET, MML version 4.99.1005

:: FRECHET:th 1
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
   rng b2 is Element of bool the carrier of b1;

:: FRECHET:th 2
theorem
for b1 being non empty 1-sorted
for b2 being 1-sorted
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st rng b3 c= the carrier of b2
   holds b3 is Function-like quasi_total Relation of NAT,the carrier of b2;

:: FRECHET:th 3
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being Basis of b2 holds
   b3 <> {};

:: FRECHET:condreg 1
registration
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Element of the carrier of a1;
  cluster -> non empty (Basis of a2);
end;

:: FRECHET:th 4
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is open(b1) & b3 is closed(b1)
   holds b2 \ b3 is open(b1);

:: FRECHET:th 5
theorem
for b1 being TopStruct
      st {} b1 is closed(b1) &
         [#] b1 is closed(b1) &
         (for b2, b3 being Element of bool the carrier of b1
               st b2 is closed(b1) & b3 is closed(b1)
            holds b2 \/ b3 is closed(b1)) &
         (for b2 being Element of bool bool the carrier of b1
               st b2 is closed(b1)
            holds meet b2 is closed(b1))
   holds b1 is TopSpace-like TopStruct;

:: FRECHET:th 6
theorem
for b1 being TopSpace-like TopStruct
for b2 being non empty TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st for b4 being Element of bool the carrier of b2 holds
              b4 is closed(b2)
           iff
              b3 " b4 is closed(b1)
   holds b2 is TopSpace-like TopStruct;

:: FRECHET:th 7
theorem
for b1 being Element of the carrier of RealSpace
for b2, b3 being Element of REAL
      st b2 = b1 & 0 < b3
   holds Ball(b1,b3) = ].b2 - b3,b2 + b3.[;

:: FRECHET:th 8
theorem
for b1 being Element of bool the carrier of R^1 holds
      b1 is open(R^1)
   iff
      for b2 being Element of REAL
            st b2 in b1
         holds ex b3 being Element of REAL st
            0 < b3 & ].b2 - b3,b2 + b3.[ c= b1;

:: FRECHET:th 9
theorem
for b1 being Function-like quasi_total Relation of NAT,the carrier of R^1
      st for b2 being Element of NAT holds
           b1 . b2 in ].b2 - (1 / 4),b2 + (1 / 4).[
   holds rng b1 is closed(R^1);

:: FRECHET:th 10
theorem
for b1 being Element of bool the carrier of R^1
      st b1 = NAT
   holds b1 is closed(R^1);

:: FRECHET:th 11
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of the carrier of TopSpaceMetr b1
for b3 being Element of the carrier of b1
      st b2 = b3
   holds ex b4 being Basis of b2 st
      b4 = {Ball(b3,1 / b5) where b5 is Element of NAT: b5 <> 0} &
       b4 is countable &
       (ex b5 being Function-like quasi_total Relation of NAT,b4 st
          for b6 being set
                st b6 in NAT
             holds ex b7 being Element of NAT st
                b6 = b7 &
                 b5 . b6 = Ball(b3,1 / (b7 + 1)));

:: FRECHET:th 12
theorem
for b1, b2 being Relation-like Function-like set holds
proj2 (b1 +* b2) = (b1 .: ((proj1 b1) \ proj1 b2)) \/ proj2 b2;

:: FRECHET:th 13
theorem
for b1, b2 being set
      st b2 c= b1
   holds (id b1) .: b2 = b2;

:: FRECHET:th 15
theorem
for b1, b2, b3 being set holds
proj1 ((id b1) +* (b2 --> b3)) = b1 \/ b2;

:: FRECHET:th 16
theorem
for b1, b2, b3 being set
      st b2 <> {}
   holds proj2 ((id b1) +* (b2 --> b3)) = (b1 \ b2) \/ {b3};

:: FRECHET:th 17
theorem
for b1, b2, b3, b4 being set
      st b3 c= b1
   holds ((id b1) +* (b2 --> b4)) " (b3 \ {b4}) = (b3 \ b2) \ {b4};

:: FRECHET:th 18
theorem
for b1, b2, b3 being set
      st not b3 in b1
   holds ((id b1) +* (b2 --> b3)) " {b3} = b2;

:: FRECHET:th 19
theorem
for b1, b2, b3, b4 being set
      st b3 c= b1 & not b4 in b1
   holds ((id b1) +* (b2 --> b4)) " (b3 \/ {b4}) = b3 \/ b2;

:: FRECHET:th 20
theorem
for b1, b2, b3, b4 being set
      st b3 c= b1 & not b4 in b1
   holds ((id b1) +* (b2 --> b4)) " (b3 \ {b4}) = b3 \ b2;

:: FRECHET:attrnot 1 => FRECHET:attr 1
definition
  let a1 be non empty TopStruct;
  attr a1 is first-countable means
    for b1 being Element of the carrier of a1 holds
       ex b2 being Basis of b1 st
          b2 is countable;
end;

:: FRECHET:dfs 1
definiens
  let a1 be non empty TopStruct;
To prove
     a1 is first-countable
it is sufficient to prove
  thus for b1 being Element of the carrier of a1 holds
       ex b2 being Basis of b1 st
          b2 is countable;

:: FRECHET:def 1
theorem
for b1 being non empty TopStruct holds
      b1 is first-countable
   iff
      for b2 being Element of the carrier of b1 holds
         ex b3 being Basis of b2 st
            b3 is countable;

:: FRECHET:th 21
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct holds
   TopSpaceMetr b1 is first-countable;

:: FRECHET:th 22
theorem
R^1 is first-countable;

:: FRECHET:funcreg 1
registration
  cluster R^1 -> strict TopSpace-like first-countable;
end;

:: FRECHET:prednot 1 => FRECHET: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 A2 is_convergent_to A3 means
    for b1 being Element of bool the carrier of a1
          st b1 is open(a1) & a3 in b1
       holds ex b2 being Element of NAT st
          for b3 being Element of NAT
                st b2 <= b3
             holds a2 . b3 in b1;
end;

:: FRECHET:dfs 2
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
     a2 is_convergent_to a3
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a1
          st b1 is open(a1) & a3 in b1
       holds ex b2 being Element of NAT st
          for b3 being Element of NAT
                st b2 <= b3
             holds a2 . b3 in b1;

:: FRECHET:def 2
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
      b2 is_convergent_to b3
   iff
      for b4 being Element of bool the carrier of b1
            st b4 is open(b1) & b3 in b4
         holds ex b5 being Element of NAT st
            for b6 being Element of NAT
                  st b5 <= b6
               holds b2 . b6 in b4;

:: FRECHET:th 23
theorem
for b1 being non empty TopStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b3 = NAT --> b2
   holds b3 is_convergent_to b2;

:: FRECHET:attrnot 2 => FRECHET:attr 2
definition
  let a1 be TopStruct;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  attr a2 is convergent means
    ex b1 being Element of the carrier of a1 st
       a2 is_convergent_to b1;
end;

:: FRECHET:dfs 3
definiens
  let a1 be TopStruct;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
     a2 is convergent
it is sufficient to prove
  thus ex b1 being Element of the carrier of a1 st
       a2 is_convergent_to b1;

:: FRECHET:def 3
theorem
for b1 being TopStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b2 is convergent(b1)
   iff
      ex b3 being Element of the carrier of b1 st
         b2 is_convergent_to b3;

:: FRECHET:funcnot 1 => FRECHET:func 1
definition
  let a1 be non empty TopStruct;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  func Lim A2 -> Element of bool the carrier of a1 means
    for b1 being Element of the carrier of a1 holds
          b1 in it
       iff
          a2 is_convergent_to b1;
end;

:: FRECHET:def 4
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 bool the carrier of b1 holds
      b3 = Lim b2
   iff
      for b4 being Element of the carrier of b1 holds
            b4 in b3
         iff
            b2 is_convergent_to b4;

:: FRECHET:attrnot 3 => FRECHET:attr 3
definition
  let a1 be non empty TopStruct;
  attr a1 is Frechet means
    for b1 being Element of bool the carrier of a1
    for b2 being Element of the carrier of a1
          st b2 in Cl b1
       holds ex b3 being Function-like quasi_total Relation of NAT,the carrier of a1 st
          rng b3 c= b1 & b2 in Lim b3;
end;

:: FRECHET:dfs 5
definiens
  let a1 be non empty TopStruct;
To prove
     a1 is Frechet
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a1
    for b2 being Element of the carrier of a1
          st b2 in Cl b1
       holds ex b3 being Function-like quasi_total Relation of NAT,the carrier of a1 st
          rng b3 c= b1 & b2 in Lim b3;

:: FRECHET:def 5
theorem
for b1 being non empty TopStruct holds
      b1 is Frechet
   iff
      for b2 being Element of bool the carrier of b1
      for b3 being Element of the carrier of b1
            st b3 in Cl b2
         holds ex b4 being Function-like quasi_total Relation of NAT,the carrier of b1 st
            rng b4 c= b2 & b3 in Lim b4;

:: FRECHET:attrnot 4 => FRECHET:attr 4
definition
  let a1 be non empty TopStruct;
  attr a1 is sequential means
    for b1 being Element of bool the carrier of a1 holds
          b1 is closed(a1)
       iff
          for b2 being Function-like quasi_total Relation of NAT,the carrier of a1
                st b2 is convergent(a1) & rng b2 c= b1
             holds Lim b2 c= b1;
end;

:: FRECHET:dfs 6
definiens
  let a1 be non empty TopStruct;
To prove
     a1 is sequential
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a1 holds
          b1 is closed(a1)
       iff
          for b2 being Function-like quasi_total Relation of NAT,the carrier of a1
                st b2 is convergent(a1) & rng b2 c= b1
             holds Lim b2 c= b1;

:: FRECHET:def 6
theorem
for b1 being non empty TopStruct holds
      b1 is sequential
   iff
      for b2 being Element of bool the carrier of b1 holds
            b2 is closed(b1)
         iff
            for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
                  st b3 is convergent(b1) & rng b3 c= b2
               holds Lim b3 c= b2;

:: FRECHET:th 24
theorem
for b1 being non empty TopSpace-like TopStruct
      st b1 is first-countable
   holds b1 is Frechet;

:: FRECHET:condreg 2
registration
  cluster non empty TopSpace-like first-countable -> Frechet (TopStruct);
end;

:: FRECHET:th 26
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 b3 is convergent(b1) & rng b3 c= b2
   holds Lim b3 c= b2;

:: FRECHET:th 27
theorem
for b1 being non empty TopSpace-like TopStruct
      st for b2 being Element of bool the carrier of b1
              st for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
                      st b3 is convergent(b1) & rng b3 c= b2
                   holds Lim b3 c= b2
           holds b2 is closed(b1)
   holds b1 is sequential;

:: FRECHET:th 28
theorem
for b1 being non empty TopSpace-like TopStruct
      st b1 is Frechet
   holds b1 is sequential;

:: FRECHET:condreg 3
registration
  cluster non empty TopSpace-like Frechet -> sequential (TopStruct);
end;

:: FRECHET:funcnot 2 => FRECHET:func 2
definition
  func REAL? -> non empty strict TopSpace-like TopStruct means
    the carrier of it = (REAL \ NAT) \/ {REAL} &
     (ex b1 being Function-like quasi_total Relation of the carrier of R^1,the carrier of it st
        b1 = (id REAL) +* (NAT --> REAL) &
         (for b2 being Element of bool the carrier of it holds
               b2 is closed(it)
            iff
               b1 " b2 is closed(R^1)));
end;

:: FRECHET:def 7
theorem
for b1 being non empty strict TopSpace-like TopStruct holds
      b1 = REAL?
   iff
      the carrier of b1 = (REAL \ NAT) \/ {REAL} &
       (ex b2 being Function-like quasi_total Relation of the carrier of R^1,the carrier of b1 st
          b2 = (id REAL) +* (NAT --> REAL) &
           (for b3 being Element of bool the carrier of b1 holds
                 b3 is closed(b1)
              iff
                 b2 " b3 is closed(R^1)));

:: FRECHET:th 30
theorem
REAL is Element of the carrier of REAL?;

:: FRECHET:th 31
theorem
for b1 being Element of bool the carrier of REAL? holds
      b1 is open(REAL?) & REAL in b1
   iff
      ex b2 being Element of bool the carrier of R^1 st
         b2 is open(R^1) &
          NAT c= b2 &
          b1 = (b2 \ NAT) \/ {REAL};

:: FRECHET:th 32
theorem
for b1 being set holds
      b1 is Element of bool the carrier of REAL? & not REAL in b1
   iff
      b1 is Element of bool the carrier of R^1 & NAT /\ b1 = {};

:: FRECHET:th 33
theorem
for b1 being Element of bool the carrier of R^1
for b2 being Element of bool the carrier of REAL?
      st b1 = b2
   holds    NAT /\ b1 = {} & b1 is open(R^1)
   iff
      not REAL in b2 & b2 is open(REAL?);

:: FRECHET:th 34
theorem
for b1 being Element of bool the carrier of REAL?
      st b1 = {REAL}
   holds b1 is closed(REAL?);

:: FRECHET:th 35
theorem
REAL? is not first-countable;

:: FRECHET:th 36
theorem
REAL? is Frechet;

:: FRECHET:th 37
theorem
ex b1 being non empty TopSpace-like TopStruct st
   b1 is Frechet & b1 is not first-countable;

:: FRECHET:th 39
theorem
for b1, b2 being Relation-like Function-like set
      st b1 tolerates b2
   holds proj2 (b1 +* b2) = (proj2 b1) \/ proj2 b2;

:: FRECHET:th 40
theorem
for b1 being Element of REAL
      st 0 < b1
   holds ex b2 being Element of NAT st
      1 / b2 < b1 & 0 < b2;