 reserve x, y, y1, y2 for set;
 reserve V for Z_Module;
 reserve u, v, w for Vector of V;
 reserve F, G, H, I for FinSequence of V;
 reserve W, W1, W2, W3 for Submodule of V;
 reserve KL1, KL2 for Linear_Combination of V;
 reserve X for Subset of V;

theorem Th16:
  for A being Subset of V st A is linearly-independent &
  A c= the carrier of W holds A is linearly-independent Subset of W
  proof
    let A be Subset of V such that
    A1: A is linearly-independent and
    A2: A c= the carrier of W;
    reconsider A9 = A as Subset of W by A2;
    now
      assume A9 is linearly-dependent;
      then consider K being Linear_Combination of A9 such that
      A3: Sum(K) = 0.W and
      A4: Carrier(K) <> {};
      consider L being Linear_Combination of V such that
      A5: Carrier(L) = Carrier(K) and
      A6: Sum(L) = Sum(K) by Th12;
      reconsider L as Linear_Combination of A by A5,VECTSP_6:def 4;
      Sum(L) = 0.V by A3,A6,ZMODUL01:26;
      hence contradiction by A1,A4,A5;
    end;
    hence thesis;
  end;
