 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 Th10:
  for L being Linear_Combination of V for A being Subset of V st
  Carrier(L) c= the carrier of Lin(A) holds
  ex K being Linear_Combination of A st Sum(L) = Sum(K)
  proof
    let L be Linear_Combination of V, A be Subset of V;
    consider F being FinSequence of V such that
    F is one-to-one and
    A1: rng F = Carrier(L) and
    A2: Sum(L) = Sum(L (#) F) by VECTSP_6:def 6;
    assume Carrier(L) c= the carrier of Lin(A);
    then consider K being Linear_Combination of A such that
    A3: Sum(L (#) F) = Sum(K) by A1,Th9;
    take K;
    thus thesis by A2,A3;
  end;
