reserve V,W for Z_Module;
reserve T for linear-transformation of V,W;
reserve T for linear-transformation of V,W;
reserve l for Linear_Combination of V;
reserve V,W for Z_Module;
reserve l for Linear_Combination of V;
reserve T for linear-transformation of V,W;

theorem ThTF3C3:
  for R being Ring
  for V be LeftMod of R,
  A be finite Subset of V,
  l,l0 be Linear_Combination of V
  st l | (Carrier l0) = l0 | (Carrier l0)
  & Carrier l0 c= Carrier l & A c= Carrier l0
  holds Sum(l*canFS(A)) = Sum(l0*canFS(A))
  proof
    let R be Ring;
    let V be LeftMod of R,
    A be finite Subset of V,
    l,l0 be Linear_Combination of V;
    assume
    A1: l | (Carrier l0) = l0 | (Carrier l0)
    & Carrier l0 c= Carrier l & A c= Carrier l0;
    dom l0 = the carrier of V by FUNCT_2:def 1;
    then dom(l0 | (Carrier l0)) = (Carrier l0) by RELAT_1:62; then
    A2: rng (canFS(A)) c= dom (l0 | (Carrier l0)) by A1;
    then l * canFS(A) = (l0 | (Carrier l0)) * canFS(A) by A1,RELAT_1:165;
    hence thesis by A2,RELAT_1:165;
  end;
