
theorem
  for V being RealUnitarySpace, v being VECTOR of V, r being Real st r =
  0 holds Ball(v,r) is empty
proof
  let V be RealUnitarySpace;
  let v be VECTOR of V;
  let r be Real;
  assume
A1: r = 0;
  assume Ball(v,r) is non empty;
  then consider u being object such that
A2: u in Ball(v,r);
  u in {y where y is Point of V : ||.v - y.|| < r} by A2,BHSP_2:def 5;
  then ex w being Point of V st u = w & ||.v - w.|| < r;
  hence contradiction by A1,BHSP_1:28;
end;
