reserve V for Z_Module;
reserve W, W1, W2 for Submodule of V;

theorem
  for V being Z_Module st (Omega).V is free holds V is free
  proof
    let V be Z_Module such that
    A1: (Omega).V is free;
    consider I be Subset of (Omega).V such that
    a2: I is base by VECTSP_7:def 4,A1;
    A2: I is linearly-independent & (Omega).V = Lin(I) by a2,VECTSP_7:def 3;
    reconsider II = I as linearly-independent Subset of V by A2,ZMODUL03:15;
    (Omega).V = Lin(II) by A2,ZMODUL03:20; then
    II is base by VECTSP_7:def 3;
    hence thesis by VECTSP_7:def 4;
  end;
