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 x in WW1 & x in WW2 by A1,VECTSP_5:3;
        hence x in WW1 /\ WW2 by VECTSP_5:3;
      end;
      assume x in WW1 /\ WW2;
      then x in W1 & x in W2 by A1,VECTSP_5:3;
      hence x in W1 /\ W2 by VECTSP_5:3;
    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;
