(** We need a notion of equality over planes. *)
Definition EqP : relation Plane := fun p q => forall X, IncidentP X p <-> IncidentP X q.
Infix "=p=" := EqP (at level 70):type_scope.

Lemma incidentp_eqp : forall A B C p, forall H : ~ Col_H A B C,
IncidentP A p -> IncidentP B p -> IncidentP C p ->
(Plan A B C H) =p= p.