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
 for X being symmetric triangle non empty Reflexive MetrStruct,
     V being bounded Subset of X, x being Element of X
  ex r being Real st 0 < r & V c= Ball(x,r)
 proof let X be symmetric triangle non empty Reflexive MetrStruct,
     V be bounded Subset of X, y be Element of X;
   consider r being Real, x being Element of X such that
A1:  0 < r and
A2: V c= Ball(x,r) by Def3;
  take s = r + dist(x,y);
   dist(x,y) >= 0 by METRIC_1:5;
   then s >= r+0 by XREAL_1:7;
  hence 0 < s by A1;
  let e be object;
  assume
A3:  e in V;
   then reconsider e as Element of X;
   e in Ball(x,r) by A3,A2;
   then  dist(e,x) < r by METRIC_1:11;
   then
A4:   dist(e,x) + dist(x,y) < r + dist(x,y) by XREAL_1:8;
   dist(e,y) <= dist(e,x) + dist(x,y) by METRIC_1:4;
   then dist(e,y) < r + dist(x,y) by A4,XXREAL_0:2;
   then e in Ball(y,s) by METRIC_1:11;
  hence thesis;
 end;
