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.

(* Specifications: *)
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.
Lemma axiom_hcong_1_uniqueness :
forall A B l M A' B' A'' B'', A <> B -> IncidentL M l ->
IncidentL A' l -> IncidentL B' l ->
IncidentL A'' l -> IncidentL B'' l ->
Between_H A' M B' -> Cong M A' A B ->
Cong M B' A B -> Between_H A'' M B'' ->
Cong M A'' A B -> Cong M B'' A B ->
(A' = A'' /\ B' = B'') \/ (A' = B'' /\ B' = A'').