reserve M for non empty MetrSpace,
  c,g1,g2 for Element of M;
reserve N for non empty MetrStruct,
  w for Element of N,
  G for Subset-Family of N,
  C for Subset of N;
reserve R for Reflexive non empty MetrStruct;
reserve T for Reflexive symmetric triangle non empty MetrStruct,
  t1 for Element of T,
  Y for Subset-Family of T,
  P for Subset of T;
reserve f for Function,
  n,m,p,n1,n2,k for Nat,
  r,s,L for Real,
  x,y for set;
reserve S1 for sequence of M,
  S2 for sequence of N;

theorem
  0<r implies diameter Ball(t1,r) <= 2*r
proof
A1: for x,y being Point of T st x in Ball(t1,r) & y in Ball(t1,r) holds dist
  (x,y)<= 2*r
  proof
    let x,y be Point of T;
    assume x in Ball(t1,r) & y in Ball(t1,r);
    then dist(t1,x)<r & dist(t1,y)<r by METRIC_1:11;
    then
A2: dist(t1,x)+dist(t1,y)<r+r by XREAL_1:8;
    dist(x,y)<=dist(x,t1)+dist(t1,y) by METRIC_1:4;
    hence thesis by A2,XXREAL_0:2;
  end;
  assume 0<r;
  then t1 in Ball(t1,r) by Th11;
  hence thesis by A1,Def8;
end;
