
theorem
  for G1, G2 being _Graph, W1 being Walk of G1, W2 being Walk of G2
  holds len W1 <= len W2 iff W1.length() <= W2.length()
proof
  let G1, G2 be _Graph, W1 be Walk of G1, W2 be Walk of G2;
  hereby
    assume len W1 <= len W2;
    then 2*W1.length() + 1 <= len W2 by GLIB_001:112;
    then 2*W1.length() + 1 <= 2*W2.length() + 1 by GLIB_001:112;
    then 2*W1.length() <= 2*W2.length() by XREAL_1:6;
    then 2*W1.length()/2 <= 2*W2.length()/2 by XREAL_1:72;
    hence W1.length() <= W2.length();
  end;
  assume W1.length() <= W2.length();
  then 2*W1.length() <= 2*W2.length() by XREAL_1:64;
  then 2*W1.length() + 1 <= 2*W2.length() + 1 by XREAL_1:6;
  then len W1 <= 2*W2.length() + 1 by GLIB_001:112;
  hence thesis by GLIB_001:112;
end;
