 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
  for A being Subset of W st A is linearly-independent holds
  A is linearly-independent Subset of V
  proof
    let A be Subset of W;
    the carrier of W c= the carrier of V by VECTSP_4:def 2;
    then reconsider A9= A as Subset of V by XBOOLE_1:1;
    assume A1: A is linearly-independent;
    now
      assume A9 is linearly-dependent;
      then consider L being Linear_Combination of A9 such that
      A2: Sum(L) = 0.V and
      A3: Carrier(L) <> {};
      Carrier(L) c= A by VECTSP_6:def 4;
      then consider LW being Linear_Combination of W such that
      A4: Carrier(LW) = Carrier(L) and
      A5: Sum(LW) = Sum(L) by Th13,XBOOLE_1:1;
      reconsider LW as Linear_Combination of A by A4,VECTSP_6:def 4;
      Sum(LW) = 0.W by A2,A5,ZMODUL01:26;
      hence contradiction by A1,A3,A4;
    end;
    hence thesis;
  end;
