
theorem
  for V being RealUnitarySpace, v being VECTOR of V, r being Real st the
  carrier of V = {0.V} & r <> 0 holds Sphere(v,r) is empty
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;
  assume Sphere(v,r) is non empty;
  then consider x being object such that
A3: x in Sphere(v,r);
  Sphere(v,r) = {y where y is Point of V : ||.v - y.|| = r} by BHSP_2:def 7;
  then consider w being Point of V such that
  x = w and
A4: ||.v-w.|| = r by A3;
  v-w <> 0.V by A2,A4,BHSP_1:26;
  hence contradiction by A1,TARSKI:def 1;
end;
