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;
