reserve x, y, y1, y2 for object;
reserve V for Z_Module;
reserve W, W1, W2 for Submodule of V;
reserve u, v for VECTOR of V;
reserve i, j, k, n for Element of NAT;

theorem LmRankSX2:
  for V being torsion-free Z_Module,
  W1, W2 being finite-rank free Submodule of V,
  I1 being Basis of W1 holds
  ex I being finite linearly-independent Subset of V
  st I is Subset of W1 + W2 & I1 c= I
  & rank(W1 + W2) = rank(Lin(I))
  proof
    let V be torsion-free Z_Module,
    W1, W2 be finite-rank free Submodule of V,
    I1 be Basis of W1;
    A2: W1 is Submodule of W1 + W2 by ZMODUL01:97;
    then I1 c= the carrier of W1 & the carrier of W1 c= the carrier of W1 + W2
    by VECTSP_4:def 2;
    then I1 c= the carrier of W1 + W2;
    then reconsider II1 = I1 as Subset of W1 + W2;
    reconsider II1 as finite Subset of W1 + W2;
    reconsider II1 as finite linearly-independent Subset of W1 + W2
    by A2,VECTSP_7:def 3,ZMODUL03:15;
    consider II be finite linearly-independent Subset of W1 + W2 such that
    A3: II1 c= II & rank(W1 + W2) = card(II) by LmRankSX1;
    II c= the carrier of W1 + W2 & the carrier of W1 + W2 c= the carrier of V
    by VECTSP_4:def 2;
    then reconsider I = II as Subset of V by XBOOLE_1:1;
    reconsider I as finite linearly-independent Subset of V by ZMODUL03:15;
    rank(W1 + W2) = rank(Lin(I)) by A3,ZMODUL05:3;
    hence thesis by A3;
  end;
