 reserve V for Z_Module;
 reserve W for Subspace of V;
 reserve v, u for Vector of V;
 reserve i for Element of INT.Ring;

theorem LMThSumMod2:
  for R being Ring
  for V being LeftMod of R, W being Subspace of V,
  W1s, W2s being Subspace of W,
  W1, W2 being Subspace of V
  st W1s = W1 & W2s = W2
  holds W1s + W2s = W1 + W2
  proof
    let R be Ring;
    let V be LeftMod of R, W be Subspace of V,
    W1s, W2s be Subspace of W,
    W1, W2 be Subspace of V;
    assume AS: W1s = W1 & W2s = W2;
    reconsider SWs12 = W1s + W2s as strict Subspace of V by VECTSP_4:26;
    for v be Vector of V holds v in SWs12 iff v in W1 + W2
    proof
      let v be Vector of V;
      hereby assume v in SWs12;
        then consider x1, x2 be Vector of W such that
        B1: x1 in W1s & x2 in W2s & v = x1 + x2 by VECTSP_5:1;
        the carrier of W1 c= the carrier of V by VECTSP_4:def 2;
        then reconsider xx1 = x1 as Vector of V by AS,B1;
        the carrier of W2 c= the carrier of V by VECTSP_4:def 2;
        then reconsider xx2 = x2 as Vector of V by AS,B1;
        v = xx1 + xx2 by VECTSP_4:13,B1;
        hence v in W1 + W2 by VECTSP_5:1,AS,B1;
      end;
      assume v in W1 + W2;
      then consider x1, x2 be Vector of V such that
      B1: x1 in W1 & x2 in W2 & v = x1 + x2 by VECTSP_5:1;
      the carrier of W1s c= the carrier of W by VECTSP_4:def 2;
      then reconsider xx1 = x1 as Vector of W by AS,B1;
      the carrier of W2s c= the carrier of W by VECTSP_4:def 2;
      then reconsider xx2 = x2 as Vector of W by AS,B1;
      v = xx1 + xx2 by VECTSP_4:13,B1;
      hence v in SWs12 by AS,B1,VECTSP_5:1;
    end;
    hence thesis by VECTSP_4:30;
  end;
