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