
theorem
  for V being strict RealUnitarySpace, W being strict Subspace of V st (
  for v being VECTOR of V holds v in W iff v in V) holds W = V
proof
  let V be strict RealUnitarySpace;
  let W be strict Subspace of V;
  assume
A1: for v being VECTOR of V holds v in W iff v in V;
  V is Subspace of V by Th19;
  hence thesis by A1,Th25;
end;
