theorem
  for S, V being Subset of R holds S is bounded & V c= S implies
  diameter V <= diameter S
proof
  let S, V be Subset of R;
  assume that
A1: S is bounded and
A2: V c= S;
A3: V is bounded by A1,A2,Th14;
  per cases;
  suppose
    V = {};
    then diameter V = 0 by Def8;
    hence thesis by A1,Th21;
  end;
  suppose
A4: V <> {};
    for x,y being Point of R st x in V & y in V holds dist(x,y)<=(diameter
    S ) by A1,A2,Def8;
    hence thesis by A3,A4,Def8;
  end;
end;
