 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 HM6:
  for R being Ring
  for X, Y be LeftMod of R, A be Subset of X,
      L be linear-transformation of X, Y
  st L is bijective
  holds A is linearly-independent
  iff L.:A is linearly-independent
  proof
    let R be Ring;
    let X, Y be LeftMod of R, A be Subset of X,
    L be linear-transformation of X, Y;
    assume AS1: L is bijective;
    D1: dom L = the carrier of X by FUNCT_2:def 1;
    consider K be linear-transformation of Y, X such that
    AS3: K= L" & K is bijective by HM1,AS1;
    thus A is linearly-independent implies L.:A is linearly-independent
    by AS1,HM4;
    assume L.:A is linearly-independent;
    then K.: (L.:A) is linearly-independent by AS3,HM4;
    hence thesis by D1,AS1,AS3,FUNCT_1:107;
  end;
