(* 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_left_right_invariant :
forall B P Q U V Y : Point,
inter_ll Y P Q U V -> S4 P U Q V <> 0.