Definition Qpositive_sub (w w' : Qpositive) :=
  let (p, q) := Qpositive_i w in
  let (p', q') := Qpositive_i w' in
  Qpositive_c (p * q' - p' * q) (q * q') (p * q' + p' * q + q * q').

Theorem Qpositive_sub_simpl_l : forall w w' w'' : Qpositive, w <> w'' :>Qpositive -> w' <> w'' :>Qpositive -> Qpositive_le w w'' -> Qpositive_le w' w'' -> Qpositive_sub w'' w = Qpositive_sub w'' w' -> w = w'.