Article TBSP_1, MML version 4.99.1005

:: TBSP_1:th 1
theorem
for b1 being Element of REAL
   st 0 < b1 & b1 < 1
for b2, b3 being Element of NAT
      st b2 <= b3
   holds b1 to_power b3 <= b1 to_power b2;

:: TBSP_1:th 2
theorem
for b1 being Element of REAL
   st 0 < b1 & b1 < 1
for b2 being Element of NAT holds
   b1 to_power b2 <= 1 & 0 < b1 to_power b2;

:: TBSP_1:th 3
theorem
for b1 being Element of REAL
   st 0 < b1 & b1 < 1
for b2 being Element of REAL
      st 0 < b2
   holds ex b3 being Element of NAT st
      b1 to_power b3 < b2;

:: TBSP_1:attrnot 1 => TBSP_1:attr 1
definition
  let a1 be non empty MetrStruct;
  attr a1 is totally_bounded means
    for b1 being Element of REAL
          st 0 < b1
       holds ex b2 being Element of bool bool the carrier of a1 st
          b2 is finite &
           the carrier of a1 = union b2 &
           (for b3 being Element of bool the carrier of a1
                 st b3 in b2
              holds ex b4 being Element of the carrier of a1 st
                 b3 = Ball(b4,b1));
end;

:: TBSP_1:dfs 1
definiens
  let a1 be non empty MetrStruct;
To prove
     a1 is totally_bounded
it is sufficient to prove
  thus for b1 being Element of REAL
          st 0 < b1
       holds ex b2 being Element of bool bool the carrier of a1 st
          b2 is finite &
           the carrier of a1 = union b2 &
           (for b3 being Element of bool the carrier of a1
                 st b3 in b2
              holds ex b4 being Element of the carrier of a1 st
                 b3 = Ball(b4,b1));

:: TBSP_1:def 1
theorem
for b1 being non empty MetrStruct holds
      b1 is totally_bounded
   iff
      for b2 being Element of REAL
            st 0 < b2
         holds ex b3 being Element of bool bool the carrier of b1 st
            b3 is finite &
             the carrier of b1 = union b3 &
             (for b4 being Element of bool the carrier of b1
                   st b4 in b3
                holds ex b5 being Element of the carrier of b1 st
                   b4 = Ball(b5,b2));

:: TBSP_1:th 5
theorem
for b1 being non empty MetrStruct
for b2 being Relation-like Function-like set holds
      b2 is Function-like quasi_total Relation of NAT,the carrier of b1
   iff
      proj1 b2 = NAT &
       (for b3 being Element of NAT holds
          b2 . b3 is Element of the carrier of b1);

:: TBSP_1:attrnot 2 => TBSP_1:attr 2
definition
  let a1 be non empty MetrStruct;
  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
       for b2 being Element of REAL
             st 0 < b2
          holds ex b3 being Element of NAT st
             for b4 being Element of NAT
                   st b3 <= b4
                holds dist(a2 . b4,b1) < b2;
end;

:: TBSP_1:dfs 2
definiens
  let a1 be non empty MetrStruct;
  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
       for b2 being Element of REAL
             st 0 < b2
          holds ex b3 being Element of NAT st
             for b4 being Element of NAT
                   st b3 <= b4
                holds dist(a2 . b4,b1) < b2;

:: TBSP_1:def 3
theorem
for b1 being non empty MetrStruct
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
         for b4 being Element of REAL
               st 0 < b4
            holds ex b5 being Element of NAT st
               for b6 being Element of NAT
                     st b5 <= b6
                  holds dist(b2 . b6,b3) < b4;

:: TBSP_1:funcnot 1 => TBSP_1:func 1
definition
  let a1 be non empty Reflexive discerning symmetric triangle MetrStruct;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  assume a2 is convergent(a1);
  func lim A2 -> Element of the carrier of a1 means
    for b1 being Element of REAL
          st 0 < b1
       holds ex b2 being Element of NAT st
          for b3 being Element of NAT
                st b2 <= b3
             holds dist(a2 . b3,it) < b1;
end;

:: TBSP_1:def 4
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
   st b2 is convergent(b1)
for b3 being Element of the carrier of b1 holds
      b3 = lim b2
   iff
      for b4 being Element of REAL
            st 0 < b4
         holds ex b5 being Element of NAT st
            for b6 being Element of NAT
                  st b5 <= b6
               holds dist(b2 . b6,b3) < b4;

:: TBSP_1:attrnot 3 => TBSP_1:attr 3
definition
  let a1 be non empty MetrStruct;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  attr a2 is Cauchy means
    for b1 being Element of REAL
          st 0 < b1
       holds ex b2 being Element of NAT st
          for b3, b4 being Element of NAT
                st b2 <= b3 & b2 <= b4
             holds dist(a2 . b3,a2 . b4) < b1;
end;

:: TBSP_1:dfs 4
definiens
  let a1 be non empty MetrStruct;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
     a2 is Cauchy
it is sufficient to prove
  thus for b1 being Element of REAL
          st 0 < b1
       holds ex b2 being Element of NAT st
          for b3, b4 being Element of NAT
                st b2 <= b3 & b2 <= b4
             holds dist(a2 . b3,a2 . b4) < b1;

:: TBSP_1:def 5
theorem
for b1 being non empty MetrStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b2 is Cauchy(b1)
   iff
      for b3 being Element of REAL
            st 0 < b3
         holds ex b4 being Element of NAT st
            for b5, b6 being Element of NAT
                  st b4 <= b5 & b4 <= b6
               holds dist(b2 . b5,b2 . b6) < b3;

:: TBSP_1:attrnot 4 => TBSP_1:attr 4
definition
  let a1 be non empty MetrStruct;
  attr a1 is complete means
    for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
          st b1 is Cauchy(a1)
       holds b1 is convergent(a1);
end;

:: TBSP_1:dfs 5
definiens
  let a1 be non empty MetrStruct;
To prove
     a1 is complete
it is sufficient to prove
  thus for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
          st b1 is Cauchy(a1)
       holds b1 is convergent(a1);

:: TBSP_1:def 6
theorem
for b1 being non empty MetrStruct holds
      b1 is complete
   iff
      for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
            st b2 is Cauchy(b1)
         holds b2 is convergent(b1);

:: TBSP_1:th 7
theorem
for b1 being non empty MetrStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b1 is triangle & b1 is symmetric & b2 is convergent(b1)
   holds b2 is Cauchy(b1);

:: TBSP_1:condreg 1
registration
  let a1 be non empty symmetric triangle MetrStruct;
  cluster Function-like quasi_total convergent -> Cauchy (Relation of NAT,the carrier of a1);
end;

:: TBSP_1:th 8
theorem
for b1 being non empty MetrStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b1 is symmetric
   holds    b2 is Cauchy(b1)
   iff
      for b3 being Element of REAL
            st 0 < b3
         holds ex b4 being Element of NAT st
            for b5, b6 being Element of NAT
                  st b4 <= b5
               holds dist(b2 . (b5 + b6),b2 . b5) < b3;

:: TBSP_1:th 9
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being contraction of b1
      st b1 is complete
   holds ex b3 being Element of the carrier of b1 st
      b2 . b3 = b3 &
       (for b4 being Element of the carrier of b1
             st b2 . b4 = b4
          holds b4 = b3);

:: TBSP_1:th 10
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct
      st TopSpaceMetr b1 is compact
   holds b1 is complete;

:: TBSP_1:th 12
theorem
for b1 being non empty MetrStruct
      st b1 is Reflexive & b1 is triangle & TopSpaceMetr b1 is compact
   holds b1 is totally_bounded;

:: TBSP_1:attrnot 5 => TBSP_1:attr 5
definition
  let a1 be non empty MetrStruct;
  attr a1 is bounded means
    ex b1 being Element of REAL st
       0 < b1 &
        (for b2, b3 being Element of the carrier of a1 holds
        dist(b2,b3) <= b1);
end;

:: TBSP_1:dfs 6
definiens
  let a1 be non empty MetrStruct;
To prove
     a1 is bounded
it is sufficient to prove
  thus ex b1 being Element of REAL st
       0 < b1 &
        (for b2, b3 being Element of the carrier of a1 holds
        dist(b2,b3) <= b1);

:: TBSP_1:def 8
theorem
for b1 being non empty MetrStruct holds
      b1 is bounded
   iff
      ex b2 being Element of REAL st
         0 < b2 &
          (for b3, b4 being Element of the carrier of b1 holds
          dist(b3,b4) <= b2);

:: TBSP_1:attrnot 6 => TBSP_1:attr 6
definition
  let a1 be non empty MetrStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is bounded means
    ex b1 being Element of REAL st
       0 < b1 &
        (for b2, b3 being Element of the carrier of a1
              st b2 in a2 & b3 in a2
           holds dist(b2,b3) <= b1);
end;

:: TBSP_1:dfs 7
definiens
  let a1 be non empty MetrStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is bounded
it is sufficient to prove
  thus ex b1 being Element of REAL st
       0 < b1 &
        (for b2, b3 being Element of the carrier of a1
              st b2 in a2 & b3 in a2
           holds dist(b2,b3) <= b1);

:: TBSP_1:def 9
theorem
for b1 being non empty MetrStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is bounded(b1)
   iff
      ex b3 being Element of REAL st
         0 < b3 &
          (for b4, b5 being Element of the carrier of b1
                st b4 in b2 & b5 in b2
             holds dist(b4,b5) <= b3);

:: TBSP_1:funcreg 1
registration
  let a1 be non empty set;
  cluster DiscreteSpace a1 -> strict bounded;
end;

:: TBSP_1:exreg 1
registration
  cluster non empty Reflexive discerning symmetric triangle bounded MetrStruct;
end;

:: TBSP_1:th 14
theorem
for b1 being non empty MetrStruct holds
   {} b1 is bounded(b1);

:: TBSP_1:funcreg 2
registration
  let a1 be non empty MetrStruct;
  cluster {} a1 -> bounded;
end;

:: TBSP_1:exreg 2
registration
  let a1 be non empty MetrStruct;
  cluster bounded Element of bool the carrier of a1;
end;

:: TBSP_1:th 15
theorem
for b1 being non empty MetrStruct
for b2 being Element of bool the carrier of b1 holds
   (b2 <> {} & b2 is bounded(b1) implies ex b3 being Element of REAL st
       ex b4 being Element of the carrier of b1 st
          0 < b3 &
           b4 in b2 &
           (for b5 being Element of the carrier of b1
                 st b5 in b2
              holds dist(b4,b5) <= b3)) &
    (b1 is symmetric &
     b1 is triangle &
     (ex b3 being Element of REAL st
        ex b4 being Element of the carrier of b1 st
           0 < b3 &
            b4 in b2 &
            (for b5 being Element of the carrier of b1
                  st b5 in b2
               holds dist(b4,b5) <= b3)) implies b2 is bounded(b1));

:: TBSP_1:th 16
theorem
for b1 being non empty MetrStruct
for b2 being Element of the carrier of b1
for b3 being real set
      st b1 is Reflexive & 0 < b3
   holds b2 in Ball(b2,b3);

:: TBSP_1:th 17
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being real set
      st b3 <= 0
   holds Ball(b2,b3) = {};

:: TBSP_1:th 19
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being real set holds
   Ball(b2,b3) is bounded(b1);

:: TBSP_1:th 20
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is bounded(b1) & b3 is bounded(b1)
   holds b2 \/ b3 is bounded(b1);

:: TBSP_1:th 21
theorem
for b1 being non empty MetrStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is bounded(b1) & b3 c= b2
   holds b3 is bounded(b1);

:: TBSP_1:th 22
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
      st b3 = {b2}
   holds b3 is bounded(b1);

:: TBSP_1:th 23
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct
for b2 being Element of bool the carrier of b1
      st b2 is finite
   holds b2 is bounded(b1);

:: TBSP_1:condreg 2
registration
  let a1 be non empty Reflexive symmetric triangle MetrStruct;
  cluster finite -> bounded (Element of bool the carrier of a1);
end;

:: TBSP_1:exreg 3
registration
  let a1 be non empty Reflexive symmetric triangle MetrStruct;
  cluster non empty finite Element of bool the carrier of a1;
end;

:: TBSP_1:th 24
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct
for b2 being Element of bool bool the carrier of b1
      st b2 is finite &
         (for b3 being Element of bool the carrier of b1
               st b3 in b2
            holds b3 is bounded(b1))
   holds union b2 is bounded(b1);

:: TBSP_1:th 25
theorem
for b1 being non empty MetrStruct holds
      b1 is bounded
   iff
      [#] b1 is bounded(b1);

:: TBSP_1:funcreg 3
registration
  let a1 be non empty bounded MetrStruct;
  cluster [#] a1 -> bounded;
end;

:: TBSP_1:th 26
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct
      st b1 is totally_bounded
   holds b1 is bounded;

:: TBSP_1:funcnot 2 => TBSP_1:func 2
definition
  let a1 be non empty Reflexive MetrStruct;
  let a2 be Element of bool the carrier of a1;
  assume a2 is bounded(a1);
  func diameter A2 -> Element of REAL means
    (for b1, b2 being Element of the carrier of a1
           st b1 in a2 & b2 in a2
        holds dist(b1,b2) <= it) &
     (for b1 being Element of REAL
           st for b2, b3 being Element of the carrier of a1
                   st b2 in a2 & b3 in a2
                holds dist(b2,b3) <= b1
        holds it <= b1)
    if a2 <> {}
    otherwise it = 0;
end;

:: TBSP_1:def 10
theorem
for b1 being non empty Reflexive MetrStruct
for b2 being Element of bool the carrier of b1
   st b2 is bounded(b1)
for b3 being Element of REAL holds
   (b2 = {} or    (b3 = diameter b2
    iff
       (for b4, b5 being Element of the carrier of b1
              st b4 in b2 & b5 in b2
           holds dist(b4,b5) <= b3) &
        (for b4 being Element of REAL
              st for b5, b6 being Element of the carrier of b1
                      st b5 in b2 & b6 in b2
                   holds dist(b5,b6) <= b4
           holds b3 <= b4))) &
    (b2 = {} implies    (b3 = diameter b2
    iff
       b3 = 0));

:: TBSP_1:th 28
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct
for b2 being set
for b3 being Element of bool the carrier of b1
      st b3 = {b2}
   holds diameter b3 = 0;

:: TBSP_1:th 29
theorem
for b1 being non empty Reflexive MetrStruct
for b2 being Element of bool the carrier of b1
      st b2 is bounded(b1)
   holds 0 <= diameter b2;

:: TBSP_1:th 30
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of bool the carrier of b1
      st b2 <> {} & b2 is bounded(b1) & diameter b2 = 0
   holds ex b3 being Element of the carrier of b1 st
      b2 = {b3};

:: TBSP_1:th 31
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being Element of REAL
      st 0 < b3
   holds diameter Ball(b2,b3) <= 2 * b3;

:: TBSP_1:th 32
theorem
for b1 being non empty Reflexive MetrStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is bounded(b1) & b3 c= b2
   holds diameter b3 <= diameter b2;

:: TBSP_1:th 33
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is bounded(b1) & b3 is bounded(b1) & b2 meets b3
   holds diameter (b2 \/ b3) <= (diameter b2) + diameter b3;

:: TBSP_1:th 34
theorem
for b1 being non empty Reflexive symmetric triangle MetrStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is Cauchy(b1)
   holds rng b2 is bounded(b1);