 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
  for V being Z_Module, W being free Subspace of V holds
  ex A being Subset of V st A is Subset of W &
  A is linearly-independent & Lin(A) = (Omega).W
  proof
    let V be Z_Module, W be free Subspace of V;
    consider AW be Subset of W such that
    a1: AW is base by VECTSP_7:def 4;
    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 Subset of V;
    Lin(A) = (Omega).W by a1,ZMODUL03:20;
    hence thesis by a1,ZMODUL03:15;
  end;
