Definition Py4 A B C D := Py A B D - Py C B D.Definition per A B C := Py A B C = 0.Definition perp A B C D := Py4 A C B D = 0.Definition on_foot (Y P U V : Point) := perp Y P U V /\ Col Y U V /\ U<>V.(* We introduce three additionnal axioms *)

Axiom perp_perp_para : forall A B C D E F,
 C<>D ->
 perp A B C D -> perp E F C D -> parallel A B E F.Axiom perp_para_perp : forall A B C D E F, 
  A<>B ->
  perp A B C D -> parallel A B E F -> perp E F C D.

Lemma pyth_simpl_4 : forall A B, Py A B A = Py B A B.