reserve V for Z_Module;
reserve W, W1, W2 for Submodule of V;

theorem
  for R being Ring
  for V being LeftMod of R, W1, W2 being Subspace of V,
  W1s, W2s being strict Subspace of V st W1s = (Omega).W1 & W2s = (Omega).W2
  holds W1s /\ W2s = W1 /\ W2
  proof
    let R be Ring;
    let V be LeftMod of R, W1, W2 be Subspace of V,
    W1s, W2s be strict Subspace of V such that
    A1: W1s = (Omega).W1 & W2s = (Omega).W2;
    for x be Vector of V holds x in W1 /\ W2 iff x in W1s /\ W2s
    proof
      let x be Vector of V;
      hereby
        assume x in W1 /\ W2; then
        B1: x in W1 & x in W2 by VECTSP_5:3;
        B2: x in W1s by A1,B1;
        x in W2s by A1,B1;
        hence x in W1s /\ W2s by B2,VECTSP_5:3;
      end;
      assume B1: x in W1s /\ W2s;
      x in the ModuleStr of W1 by A1,B1,VECTSP_5:3;
      then B2: x in W1;
      x in the ModuleStr of W2 by A1,B1,VECTSP_5:3;
      then x in W2;
      hence x in W1 /\ W2 by B2,VECTSP_5:3;
    end;
    hence W1 /\ W2 = W1s /\ W2s by VECTSP_4:30;
  end;
