reserve x, y, y1, y2 for object;
reserve V for Z_Module;
reserve W, W1, W2 for Submodule of V;
reserve u, v for VECTOR of V;
reserve i, j, k, n for Element of NAT;
reserve V,W for finite-rank free Z_Module;
reserve T for linear-transformation of V,W;

theorem
  for V, W being finite-rank free Z_Module,
  T being linear-transformation of V, W
  st T is one-to-one holds rank V = rank T
  proof
    let V, W be finite-rank free Z_Module,
    T be linear-transformation of V,W;
    assume T is one-to-one;
    then ker T = (0).V by ZMODUL05:25; then
    A1: nullity(T) = 0 by ZMODUL05:26;
    rank V = rank(T) + nullity(T) by Th44
    .= rank(T) by A1;
    hence thesis;
  end;
