(* shows that we need A<>B in elim_py_on_line_d_middle Lemma essai : forall A P Q Y : Point, forall r:F, r<> 0 -> on_line_d Y P Q r -> False. *)

Theorem elim_py_inter_ll_middle :
forall A B P Q U V Y : Point,
inter_ll Y P Q U V ->
Py A Y B = - (S P V U / S4 P U Q V) * Py A Q B + - (S Q U V / S4 P U Q V) * Py A P B -
- (S P V U / S4 P U Q V) * - (S Q U V / S4 P U Q V) * Py P Q P.