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 Th42:
  for R being Ring
  for V,W be LeftMod of R, X be Subset of V,
      T be linear-transformation of V,W
  for X being Subset of V,
      l be Linear_Combination of T.:X,
  v being Element of V st v in X & T|X is one-to-one holds (T#l).v = l.(T.v)
  proof
    let R be Ring;
    let V,W be LeftMod of R, X be Subset of V,
        T be linear-transformation of V,W;
    let X be Subset of V,
        l be Linear_Combination of T.:X;
    let v be Element of V;
    assume v in X & T|X is one-to-one;
    then not v in dom ((X`) --> (0.R)) &
    T#l = (l*T) +* ((X`) --> (0.R)) by Def6,XBOOLE_0:def 5; then
    A1: (T#l).v = (l*T).v by FUNCT_4:11;
    dom T = [#]V by RANKNULL:7;
    hence thesis by A1,FUNCT_1:13;
  end;
