(** Saccheri's three hypotheses *)

Definition hypothesis_of_right_saccheri_quadrilaterals := forall A B C D, Saccheri A B C D -> Per A B C.Definition hypothesis_of_acute_saccheri_quadrilaterals := forall A B C D, Saccheri A B C D -> Acute A B C.Definition hypothesis_of_obtuse_saccheri_quadrilaterals := forall A B C D, Saccheri A B C D -> Obtuse A B C.

Lemma t22_14__sams_nbet_aux : forall A B C D E F P Q R, hypothesis_of_acute_saccheri_quadrilaterals -> ~ Col A B C -> SumA C A B A B C D E F -> SumA D E F B C A P Q R -> Acute A B C -> Acute A C B -> SAMS D E F B C A /\ ~ Bet P Q R.