(** This following result is very important, it shows the existence of a midpoint. The proof is involved because we are not using continuity axioms. *)
(** This corresponds to l8_22 in Tarski's book. *)

Lemma l8_22_bis : forall A B P R X,
A <> B -> A <> P ->
Perp A B P A -> Perp A B R B ->
Cong A P B R -> Col A B X -> Bet P X R ->
Cong A R P B /\ Midpoint X A B /\ Midpoint X P R.