 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, T be linear-transformation of X, Y
  st T is bijective holds
  X is free iff Y is free
  proof
    let R be Ring;
    let X, Y be LeftMod of R, T be linear-transformation of X, Y;
    assume AS1: T is bijective;
    consider K be linear-transformation of Y, X such that
    AS3: K= T" & K is bijective by HM1,AS1;
    thus thesis by HM9,AS1,AS3;
  end;
