
theorem
  for V being RealUnitarySpace, W1,W2 being Subspace of V st (for v
  being VECTOR of V st v in W1 holds v in W2) holds W1 is Subspace of W2
proof
  let V be RealUnitarySpace;
  let W1,W2 be Subspace of V;
  assume
A1: for v being VECTOR of V st v in W1 holds v in W2;
  the carrier of W1 c= the carrier of W2
  proof
    let x be object;
    assume
A2: x in the carrier of W1;
    the carrier of W1 c= the carrier of V by Def1;
    then reconsider v = x as VECTOR of V by A2;
    v in W1 by A2;
    then v in W2 by A1;
    hence thesis;
  end;
  hence thesis by Th22;
end;
