(** 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.

Lemma cut_two_sides : forall l A B, cut l A B <-> TS (P1 l) (P2 l) A B.