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 consider x1, x2 be Vector of V such that
        B1: x1 in W1 & x2 in W2 & x = x1 + x2 by VECTSP_5:1;
        B2: x1 in W1s by A1,B1;
        x2 in W2s by A1,B1;
        hence x in W1s + W2s by B1,B2,VECTSP_5:1;
      end;
      assume x in W1s + W2s;
      then consider x1, x2 be Vector of V such that
      B1: x1 in W1s & x2 in W2s & x = x1 + x2 by VECTSP_5:1;
      B2: x1 in W1 by A1,B1;
      x2 in W2 by A1,B1;
      hence x in W1 + W2 by B1,B2,VECTSP_5:1;
    end;
    hence W1 + W2 = W1s + W2s by VECTSP_4:30;
  end;
