reserve X for MetrSpace,
  x,y,z for Element of X,
  A for non empty set,
  G for Function of [:A,A:],REAL,
  f for Function,
  k,n,m,m1,m2 for Nat,
  q,r for Real;
reserve X for non empty MetrSpace,
  x,y for Element of X,
  V for Subset of X,
  S,S1,T for sequence of X,
  Nseq for increasing sequence of NAT;

theorem Th14:
  S is_convergent_in_metrspace_to x iff dist_to_point(S,x) is
  convergent & lim dist_to_point(S,x) = 0
proof
A1: S is_convergent_in_metrspace_to x implies dist_to_point(S,x) is
  convergent & lim dist_to_point(S,x) = 0
  proof
    assume
A2: S is_convergent_in_metrspace_to x;
A3: for r be Real st 0 < r
   ex m being Nat st for n being Nat st m <= n holds |.
    dist_to_point(S,x).n - 0 .| < r
    proof
      let r be Real;
      assume
A4:   0 < r;
      reconsider r as Real;
      consider m1 such that
A5:   for n st m1 <= n holds dist(S.n,x) < r by A2,A4;
      take k = m1;
      let n be Nat;
      assume k <= n;
      then dist(S.n,x) < r by A5;
      then
A6:   dist_to_point(S,x).n < r by Def6;
      dist(S.n,x) = dist_to_point(S,x).n by Def6;
      then 0 <= dist_to_point(S,x).n by METRIC_1:5;
      hence thesis by A6,ABSVALUE:def 1;
    end;
    then dist_to_point(S,x) is convergent by SEQ_2:def 6;
    hence thesis by A3,SEQ_2:def 7;
  end;
  dist_to_point(S,x) is convergent & lim dist_to_point(S,x) = 0 implies S
  is_convergent_in_metrspace_to x
  proof
    assume
A7: dist_to_point(S,x) is convergent & lim dist_to_point(S,x) = 0;
    for r st 0 < r ex m st for n st m <= n holds dist(S.n,x) < r
    proof
      let r;
      assume 0 < r;
      then consider m1 being Nat such that
A8:   for n being Nat st m1 <= n holds |.dist_to_point(S,x).n - 0 .| < r by A7,
SEQ_2:def 7;
       reconsider  k = m1 as Element of NAT by ORDINAL1:def 12;
      take k;
      let n;
      assume k <= n;
      then |.dist_to_point(S,x).n - 0 .| < r by A8;
      then
A9:   |.dist(S.n,x).| < r by Def6;
      0 <= dist(S.n,x) by METRIC_1:5;
      hence thesis by A9,ABSVALUE:def 1;
    end;
    hence thesis;
  end;
  hence thesis by A1;
end;
