 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 LmFree2:
  for V being Z_Module, W being finite-rank free Subspace of V holds
  ex A being finite Subset of V st A is finite Subset of W &
  A is linearly-independent & Lin(A) = (Omega).W & card(A) = rank(W)
  proof
    let V be Z_Module, W be finite-rank free Subspace of V;
    consider AW be finite Subset of W such that
    A1: AW is Basis of W by ZMODUL03:def 3;
    A2: AW is linearly-independent & Lin(AW) = (Omega).W by A1,VECTSP_7:def 3;
    AW c= the carrier of W & the carrier of W c= the carrier of V
    by VECTSP_4:def 2;
    then AW c= the carrier of V;
    then reconsider A = AW as finite Subset of V;
    A3: A is linearly-independent by ZMODUL03:15,A1,VECTSP_7:def 3;
    A4: rank(W) = card(A) by A1,ZMODUL03:def 5;
    Lin(A) = (Omega).W by A2,ZMODUL03:20;
    hence thesis by A3,A4;
  end;
