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,
  WW1, WW2 being Subspace of W1 + W2
  st WW1 = W1 & WW2 = W2 holds
  W1 + W2 = WW1 + WW2
  proof
    let R be Ring;
    let V be LeftMod of R, W1, W2 be Subspace of V,
    WW1, WW2 be Subspace of W1 + W2 such that
    A1: WW1 = W1 & WW2 = W2;
    A2: WW1 + WW2 is Subspace of V by VECTSP_4:26;
    for x being object holds x in W1 + W2 iff x in WW1 + WW2
    proof
      let x be object;
      hereby
        assume x in W1 + W2;
        then consider v1, v2 being Vector of V such that
        B2: v1 in W1 & v2 in W2 & x = v1 + v2 by VECTSP_5:1;
        v1 in W1 + W2 & v2 in W1 + W2 by B2,VECTSP_5:2;
        then reconsider vv1 = v1, vv2 = v2 as Vector of W1 + W2;
        v1 in WW1 & v2 in WW2 & x = vv1 + vv2 by A1,B2,VECTSP_4:13;
        hence x in WW1 + WW2 by VECTSP_5:1;
      end;
      assume x in WW1 + WW2;
      then consider vv1, vv2 being Vector of W1 + W2 such that
      B1: vv1 in WW1 & vv2 in WW2 & x = vv1 + vv2 by VECTSP_5:1;
      thus x in W1 + W2 by B1;
    end;
    then for x being Vector of V holds x in W1 + W2 iff x in WW1 + WW2;
    hence thesis by A2,VECTSP_4:30;
  end;
