theorem Th21:
  for S being Subset of R st S is bounded holds 0 <= diameter S
proof
  let S be Subset of R;
  assume
A1: S is bounded;
  per cases;
  suppose
    S = {};
    hence thesis by Def8;
  end;
  suppose
A2: S <> {};
    set x = the Element of S;
    reconsider x as Element of R by A2,TARSKI:def 3;
    dist(x,x)<=diameter S by A1,A2,Def8;
    hence thesis by METRIC_1:1;
  end;
end;
