 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 HM15:
  for X, Y be finite-rank free Z_Module, T be linear-transformation of X, Y
  st T is bijective holds
  rank(X) = rank(Y)
  proof
    let X, Y be finite-rank free Z_Module,
    T be linear-transformation of X, Y;
    assume AS1: T is bijective;
    for I being Basis of X holds rank(Y) = card I
    proof
      let I be Basis of X;
      dom T = the carrier of X by FUNCT_2:def 1; then
      X12: card I = card (T.:I) by CARD_1:5,AS1,CARD_1:33;
      T.: I is Basis of Y by AS1,HM12;
      hence rank(Y) = card I by X12,ZMODUL03:def 5;
    end;
    hence thesis by ZMODUL03:def 5;
  end;
