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 LTh29:
  for R being Ring
  for V,W be LeftMod of R,
  l be Linear_Combination of V, T be linear-transformation of V,W holds
  (T@*l) is Linear_Combination of T .: (Carrier l)
  proof
    let R be Ring;
    let V,W be LeftMod of R;
    let l be Linear_Combination of V, T be linear-transformation of V,W;
    Carrier (T@*l) c= T.:(Carrier l) by LDef5;
    hence thesis by VECTSP_6:def 4;
  end;
