(** This comes from Martin's proof in Theorem 22.10 *)

Lemma three_hypotheses_aux : forall A B C D M N A' B' C' D' M' N',
Saccheri A B C D -> Saccheri A' B' C' D' ->
Midpoint M B C -> Midpoint M' B' C' -> Midpoint N A D -> Midpoint N' A' D' ->
Le M N M' N' ->
(Per A B C <-> Per A' B' C') /\ (Acute A B C <-> Acute A' B' C').