 reserve V for Z_Module;
 reserve W for Subspace of V;
 reserve v, u for Vector of V;
 reserve i for Element of INT.Ring;

theorem
  for R being Ring
  for X, Y be LeftMod of R, X0 be Subset of X,
  L be linear-transformation of X, Y,
  l be Linear_Combination of L.:X0
  st X0 = the carrier of X & L is one-to-one
  holds L#l = l*L
  proof
    let R be Ring;
    let X, Y be LeftMod of R,
    X0 be Subset of X,
    L be linear-transformation of X, Y,
    l be Linear_Combination of L.:X0;
    assume that
    A0: X0 = the carrier of X and
    A1: L is one-to-one;
    X1: L|X0 is one-to-one by A1,FUNCT_1:52;
    X3: X0` = {} by XBOOLE_1:37,A0;
    L#l = (l*L) +* ({} --> 0.R) by X3,X1,ZMODUL05:def 9
    .= l*L;
    hence thesis;
  end;
