(* 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_on_foot_left_right :
forall B P U V Y : Point,
on_foot Y P U V ->
Py Y B Y =
(Py P U V * Py P U V * Py V B V + Py P U V * Py P V U * Py V B U +
Py P U V * Py P V U * Py U B V + Py P V U * Py P V U * Py U B U) / (Py U V U *  Py U V U).