(** Same side predicate corresponds to OS of Tarski. *)

(* Specifications: *)
Lemma same_side_one_side : forall A B l, same_side A B l -> OS (P1 l) (P2 l) A B.
Lemma one_side_same_side : forall A B l, OS (P1 l) (P2 l) A B -> same_side A B l.
Definition same_side' := fun A B X Y =>
X <> Y /\ forall l, IncidentL X l -> IncidentL Y l -> same_side A B l.
Lemma OS_distinct : forall P Q A B,
OS P Q A B -> P<>Q.
Lemma OS_same_side' :
forall P Q A B, OS P Q A B -> same_side' A B P Q.
Lemma same_side_OS :
forall P Q A B, same_side' P Q A B -> OS A B P Q.