reserve V,W for Z_Module;

theorem
  for V being finite-rank free Z_Module holds
  rank V = rank (Omega).V
  proof
    let V be finite-rank free Z_Module;
    consider I being finite Subset of V such that
    A1: I is Basis of V by ZMODUL03:def 3;
    A2: (Omega).V = Lin(I) by A1,VECTSP_7:def 3;
    card I = rank V & I is linearly-independent
    by A1,ZMODUL03:def 5,VECTSP_7:def 3;
    hence thesis by A2,RL5Th30;
  end;
