(** Axiom of Pasch, (Hilbert version). *)
(** First we define a predicate which means that the line l intersects the segment AB. *)
Definition cut := fun l A B =>
~ IncidentL A l /\ ~ IncidentL B l /\ exists I, IncidentL I l /\ Between_H A I B.
(** We show that this definition is equivalent to the predicate TS of Tarski. *)

Lemma axiom_pasch : forall A B C l p, ~ Col_H A B C ->
IncidentP A p -> IncidentP B p -> IncidentP C p -> IncidentLP l p -> ~ IncidentL C l ->
cut l A B -> cut l A C \/ cut l B C.