(** As a remark we also prove another version of this axiom as formalized in Isabelle by Phil Scott. *)
Definition same_side_scott := fun E A B => E <> A /\ E <> B /\ Col_H E A B /\ ~ Between_H A E B.
Remark axiom_hcong_scott:
forall P Q A C, A <> C -> P <> Q ->
exists B, same_side_scott A B C  /\ Cong P Q A B.
(** We define when two segments do not intersect. *)
Definition disjoint := fun A B C D => ~ exists P, Between_H A P B /\ Between_H C P D.
(** Note that two disjoint segments may share one of their extremities. *)

(* Specifications: *)
Lemma col_disjoint_bet : forall A B C, Col_H A B C -> disjoint A B B C -> Bet A B C.
Lemma axiom_hcong_3 : forall A B C A' B' C',
Col_H A B C -> Col_H A' B' C' ->
disjoint A B B C -> disjoint A' B' B' C' ->
Cong A B A' B' -> Cong B C B' C' -> Cong A C A' C'.
Lemma exists_not_incident : forall A B : Tpoint, forall  HH : A <> B , exists C, ~ IncidentL C (Lin A B HH).
Definition same_side := fun A B l => exists P, cut l A P /\ cut l B P.