
theorem
  for R being Ring
  for V being LeftMod of R, W1, W2, W3 being Subspace of V holds
  W1 /\ W2 is Subspace of (W1 + W3) /\ W2
  proof
    let R be Ring;
    let V be LeftMod of R, W1, W2, W3 be Subspace of V;
    for v being Vector of V st v in W1 /\ W2 holds v in (W1 + W3) /\ W2
    proof
      let v be Vector of V such that
      A1: v in W1 /\ W2;
      v in W1 by A1,VECTSP_5:3;
      then A2: v in W1 + W3 by VECTSP_5:2;
      v in W2 by A1,VECTSP_5:3;
      hence thesis by A2,VECTSP_5:3;
    end;
    hence thesis by VECTSP_4:28;
  end;
