Article METRIC_6, MML version 4.99.1005

:: METRIC_6:th 1
theorem
for b1 being Reflexive discerning symmetric triangle MetrStruct
for b2, b3, b4 being Element of the carrier of b1 holds
abs ((dist(b2,b3)) - dist(b4,b3)) <= dist(b2,b4);

:: METRIC_6:th 2
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of [:b1,b1:],REAL
   st b2 is_metric_of b1
for b3, b4 being Element of b1 holds
0 <= b2 .(b3,b4);

:: METRIC_6:th 3
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of [:b1,b1:],REAL holds
      b2 is_metric_of b1
   iff
      b2 is Reflexive(b1) & b2 is discerning(b1) & b2 is symmetric(b1) & b2 is triangle(b1);

:: METRIC_6:th 4
theorem
for b1 being non empty strict Reflexive discerning symmetric triangle MetrStruct holds
   the distance of b1 is Reflexive(the carrier of b1) & the distance of b1 is discerning(the carrier of b1) & the distance of b1 is symmetric(the carrier of b1) & the distance of b1 is triangle(the carrier of b1);

:: METRIC_6:th 5
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of [:b1,b1:],REAL holds
      b2 is_metric_of b1
   iff
      b2 is Reflexive(b1) &
       b2 is discerning(b1) &
       (for b3, b4, b5 being Element of b1 holds
       b2 .(b4,b5) <= (b2 .(b3,b4)) + (b2 .(b3,b5)));

:: METRIC_6:funcnot 1 => METRIC_6:func 1
definition
  let a1 be non empty set;
  let a2 be Function-like quasi_total Relation of [:a1,a1:],REAL;
  func bounded_metric(A1,A2) -> Function-like quasi_total Relation of [:a1,a1:],REAL means
    for b1, b2 being Element of a1 holds
    it .(b1,b2) = (a2 .(b1,b2)) / (1 + (a2 .(b1,b2)));
end;

:: METRIC_6:def 4
theorem
for b1 being non empty set
for b2, b3 being Function-like quasi_total Relation of [:b1,b1:],REAL holds
   b3 = bounded_metric(b1,b2)
iff
   for b4, b5 being Element of b1 holds
   b3 .(b4,b5) = (b2 .(b4,b5)) / (1 + (b2 .(b4,b5)));

:: METRIC_6:th 6
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of [:b1,b1:],REAL
      st b2 is_metric_of b1
   holds bounded_metric(b1,b2) is_metric_of b1;

:: METRIC_6:th 10
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of the carrier of b1 holds
   ex b3 being Function-like quasi_total Relation of NAT,the carrier of b1 st
      proj2 b3 = {b2};

:: METRIC_6:prednot 1 => METRIC_6:pred 1
definition
  let a1 be non empty MetrStruct;
  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_in_metrspace_to A3 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,a3) < b1;
end;

:: METRIC_6:dfs 2
definiens
  let a1 be non empty MetrStruct;
  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_in_metrspace_to a3
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 being Element of NAT
                st b2 <= b3
             holds dist(a2 . b3,a3) < b1;

:: METRIC_6:def 8
theorem
for b1 being non empty MetrStruct
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_in_metrspace_to b3
   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;

:: METRIC_6:attrnot 1 => 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
       ex b2 being Element of the carrier of a1 st
          0 < b1 & a2 c= Ball(b2,b1);
end;

:: METRIC_6:dfs 3
definiens
  let a1 be non empty symmetric triangle 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
       ex b2 being Element of the carrier of a1 st
          0 < b1 & a2 c= Ball(b2,b1);

:: METRIC_6:def 10
theorem
for b1 being non empty symmetric triangle MetrStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is bounded(b1)
   iff
      ex b3 being Element of REAL st
         ex b4 being Element of the carrier of b1 st
            0 < b3 & b2 c= Ball(b4,b3);

:: METRIC_6:attrnot 2 => METRIC_6:attr 1
definition
  let a1 be non empty MetrStruct;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  attr a2 is bounded means
    ex b1 being Element of REAL st
       ex b2 being Element of the carrier of a1 st
          0 < b1 & proj2 a2 c= Ball(b2,b1);
end;

:: METRIC_6: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 bounded
it is sufficient to prove
  thus ex b1 being Element of REAL st
       ex b2 being Element of the carrier of a1 st
          0 < b1 & proj2 a2 c= Ball(b2,b1);

:: METRIC_6:def 11
theorem
for b1 being non empty MetrStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b2 is bounded(b1)
   iff
      ex b3 being Element of REAL st
         ex b4 being Element of the carrier of b1 st
            0 < b3 & proj2 b2 c= Ball(b4,b3);

:: METRIC_6:prednot 2 => METRIC_6:pred 2
definition
  let a1 be non empty Reflexive discerning symmetric triangle MetrStruct;
  let a2 be Element of bool the carrier of a1;
  let a3 be Function-like quasi_total Relation of NAT,the carrier of a1;
  pred A2 contains_almost_all_sequence A3 means
    ex b1 being Element of NAT st
       for b2 being Element of NAT
             st b1 <= b2
          holds a3 . b2 in a2;
end;

:: METRIC_6:dfs 5
definiens
  let a1 be non empty Reflexive discerning symmetric triangle MetrStruct;
  let a2 be Element of bool the carrier of a1;
  let a3 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
     a2 contains_almost_all_sequence a3
it is sufficient to prove
  thus ex b1 being Element of NAT st
       for b2 being Element of NAT
             st b1 <= b2
          holds a3 . b2 in a2;

:: METRIC_6:def 12
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of bool the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b2 contains_almost_all_sequence b3
   iff
      ex b4 being Element of NAT st
         for b5 being Element of NAT
               st b4 <= b5
            holds b3 . b5 in b2;

:: METRIC_6:th 20
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 holds
      b2 is bounded(b1)
   iff
      ex b3 being Element of REAL st
         ex b4 being Element of the carrier of b1 st
            0 < b3 &
             (for b5 being Element of NAT holds
                b2 . b5 in Ball(b4,b3));

:: METRIC_6:th 21
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
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 is_convergent_in_metrspace_to b2
   holds b3 is convergent(b1);

:: METRIC_6:th 22
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)
   holds ex b3 being Element of the carrier of b1 st
      b2 is_convergent_in_metrspace_to b3;

:: METRIC_6:funcnot 2 => METRIC_6:func 2
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;
  let a3 be Element of the carrier of a1;
  func dist_to_point(A2,A3) -> Function-like quasi_total Relation of NAT,REAL means
    for b1 being Element of NAT holds
       it . b1 = dist(a2 . b1,a3);
end;

:: METRIC_6:def 14
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,REAL holds
      b4 = dist_to_point(b2,b3)
   iff
      for b5 being Element of NAT holds
         b4 . b5 = dist(b2 . b5,b3);

:: METRIC_6:funcnot 3 => METRIC_6:func 3
definition
  let a1 be non empty Reflexive discerning symmetric triangle MetrStruct;
  let a2, a3 be Function-like quasi_total Relation of NAT,the carrier of a1;
  func sequence_of_dist(A2,A3) -> Function-like quasi_total Relation of NAT,REAL means
    for b1 being Element of NAT holds
       it . b1 = dist(a2 . b1,a3 . b1);
end;

:: METRIC_6:def 15
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
for b4 being Function-like quasi_total Relation of NAT,REAL holds
      b4 = sequence_of_dist(b2,b3)
   iff
      for b5 being Element of NAT holds
         b4 . b5 = dist(b2 . b5,b3 . b5);

:: METRIC_6:th 26
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
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 is_convergent_in_metrspace_to b2
   holds lim b3 = b2;

:: METRIC_6:th 27
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b3 is_convergent_in_metrspace_to b2
   iff
      b3 is convergent(b1) & lim b3 = b2;

:: METRIC_6:th 28
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)
   holds ex b3 being Element of the carrier of b1 st
      b2 is_convergent_in_metrspace_to b3 & lim b2 = b3;

:: METRIC_6:th 29
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b3 is_convergent_in_metrspace_to b2
   iff
      dist_to_point(b3,b2) is convergent & lim dist_to_point(b3,b2) = 0;

:: METRIC_6:th 30
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
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 is_convergent_in_metrspace_to b2
for b4 being Element of REAL
      st 0 < b4
   holds Ball(b2,b4) contains_almost_all_sequence b3;

:: METRIC_6:th 31
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
   st for b4 being Element of REAL
           st 0 < b4
        holds Ball(b2,b4) contains_almost_all_sequence b3
for b4 being Element of bool the carrier of b1
      st b2 in b4 & b4 in Family_open_set b1
   holds b4 contains_almost_all_sequence b3;

:: METRIC_6:th 32
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st for b4 being Element of bool the carrier of b1
              st b2 in b4 & b4 in Family_open_set b1
           holds b4 contains_almost_all_sequence b3
   holds b3 is_convergent_in_metrspace_to b2;

:: METRIC_6:th 33
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b3 is_convergent_in_metrspace_to b2
   iff
      for b4 being Element of REAL
            st 0 < b4
         holds Ball(b2,b4) contains_almost_all_sequence b3;

:: METRIC_6:th 34
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b3 is_convergent_in_metrspace_to b2
   iff
      for b4 being Element of bool the carrier of b1
            st b2 in b4 & b4 in Family_open_set b1
         holds b4 contains_almost_all_sequence b3;

:: METRIC_6:th 35
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      for b4 being Element of REAL
            st 0 < b4
         holds Ball(b2,b4) contains_almost_all_sequence b3
   iff
      for b4 being Element of bool the carrier of b1
            st b2 in b4 & b4 in Family_open_set b1
         holds b4 contains_almost_all_sequence b3;

:: METRIC_6:th 36
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is convergent(b1) & b3 is convergent(b1)
   holds dist(lim b2,lim b3) = lim sequence_of_dist(b2,b3);

:: METRIC_6:th 37
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2, b3 being Element of the carrier of b1
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b4 is_convergent_in_metrspace_to b2 & b4 is_convergent_in_metrspace_to b3
   holds b2 = b3;

:: METRIC_6:th 38
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 constant
   holds b2 is convergent(b1);

:: METRIC_6:th 39
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of the carrier of b1
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b3 is_convergent_in_metrspace_to b2 & b4 is subsequence of b3
   holds b4 is_convergent_in_metrspace_to b2;

:: METRIC_6:th 40
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is Cauchy(b1) & b3 is subsequence of b2
   holds b3 is Cauchy(b1);

:: METRIC_6:th 42
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 constant
   holds b2 is Cauchy(b1);

:: METRIC_6:th 43
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)
   holds b2 is bounded(b1);

:: METRIC_6:th 44
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 Cauchy(b1)
   holds b2 is bounded(b1);

:: METRIC_6:condreg 1
registration
  let a1 be non empty Reflexive discerning symmetric triangle MetrStruct;
  cluster Function-like constant quasi_total -> convergent (Relation of NAT,the carrier of a1);
end;

:: METRIC_6:condreg 2
registration
  let a1 be non empty Reflexive discerning symmetric triangle MetrStruct;
  cluster Function-like quasi_total Cauchy -> bounded (Relation of NAT,the carrier of a1);
end;

:: METRIC_6:exreg 1
registration
  let a1 be non empty Reflexive discerning symmetric triangle MetrStruct;
  cluster Relation-like Function-like constant quasi_total convergent Cauchy bounded Relation of NAT,the carrier of a1;
end;