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 Th39:
  for R being Ring
  for V,W being LeftMod of R
  for l being Linear_Combination of V,
      T being linear-transformation of V,W holds
  T | (Carrier l) is one-to-one implies T .: (Carrier l) = Carrier(T@*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;
    assume
    A1: T | (Carrier l) is one-to-one;
    A2: T .: (Carrier l) c= Carrier(T@*l)
    proof
      let w be object;
      assume w in T .: (Carrier l);
      then consider v being object such that
      A3: v in dom T and
      A4: v in Carrier l and
      A5: T.v = w by FUNCT_1:def 6;
      reconsider v as Element of V by A3;
      (T@*l).(T.v) = l.v & l.v <> 0.R by A1,A4,Th37,ZMODUL02:8;
      hence thesis by A5;
    end;
    thus thesis by LDef5,A2;
  end;
