unfold outH.
intros.
spliter.
induction H1.
induction (eq_dec_points A B).
right; right.
split; auto.
left.
unfold Between_H.
repeat split; auto.
induction (eq_dec_points A B).
right; right.
split; auto.
right; left.
unfold Between_H.
repeat split; auto.
Qed.

Lemma axiom_hcong_1_existence : forall A B A' P l,
A <> B -> A' <> P ->
IncidentL A' l -> IncidentL P l ->
exists B', IncidentL B' l /\ outH A' P B' /\ Cong A' B' A B.