 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 V st A c= the carrier of W holds
  Lin(A) is Submodule of W
  proof
    let A be Subset of V;
    assume A1: A c= the carrier of W;
    now
      let w be object;
      assume w in the carrier of Lin(A);
      then consider L being Linear_Combination of A such that
      A2: w = Sum(L) by STRUCT_0:def 5,ZMODUL02:64;
      Carrier(L) c= A by VECTSP_6:def 4;
      then
      ex K being Linear_Combination of W st Carrier(K) = Carrier (L) & Sum(L)
      = Sum(K) by A1,Th13,XBOOLE_1:1;
      hence w in the carrier of W by A2;
    end;
    hence thesis by TARSKI:def 3,ZMODUL01:43;
  end;
