
theorem
  for V being RealUnitarySpace, v being VECTOR of V, r being Real st the
  carrier of V = {0.V} & r > 0 holds Ball(v,r) = {0.V}
proof
  let V be RealUnitarySpace;
  let v be VECTOR of V;
  let r be Real;
  assume that
A1: the carrier of V = {0.V} and
A2: r > 0;
  for w being VECTOR of V st w in {0.V} holds w in Ball(v,r)
  proof
    let w be VECTOR of V;
    assume
A3: w in {0.V};
    v = 0.V by A1,TARSKI:def 1;
    then ||.v-w.|| = ||. 0.V-0.V .|| by A3,TARSKI:def 1
      .= ||. 0.V .||
      .= 0 by BHSP_1:26;
    then w in {y where y is Point of V : ||.v - y.|| < r} by A2;
    hence thesis by BHSP_2:def 5;
  end;
  then {0.V} c= Ball(v,r);
  hence thesis by A1;
end;
