 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 Th13:
  for L being Linear_Combination of V st
  Carrier(L) c= the carrier of W
  ex K being Linear_Combination of W
   st Carrier(K) = Carrier(L) & Sum(K) = Sum (L)
  proof
    let L be Linear_Combination of V;
    assume A1: Carrier(L) c= the carrier of W;
    then reconsider C = Carrier(L) as finite Subset of W;
    the carrier of W c= the carrier of V by VECTSP_4:def 2;
    then reconsider K = L|the carrier of W as
      Function of the carrier of W, the carrier of INT.Ring by FUNCT_2:32;
    A2: K is Element of Funcs(the carrier of W, the carrier of INT.Ring)
      by FUNCT_2:8;
    A3: dom K = the carrier of W by FUNCT_2:def 1;
    now
      let w be Vector of W;
      A4: w is Vector of V by ZMODUL01:25;
      assume not w in C;
      then L.w = 0 by A4;
      hence K.w = 0 by A3,FUNCT_1:47;
    end;
    then reconsider K as Linear_Combination of W by A2,VECTSP_6:def 1;
    take K;
    thus thesis by A1,Th11;
  end;
