(** The following development is inspired by The Foundations of Geometry and the Non-Euclidean Plane, by George E Martin, chapter 22 *)

(** Additivity of the defect : if C1 is between A and C, the defect of ABC is the sum of the defects of ABC1 and BC1C. In this proof, we have to exclude the semi-elliptical case so that the sums of angles behave. *)
Lemma t22_16_1 :
~ hypothesis_of_obtuse_saccheri_quadrilaterals ->
forall A B C C1 D E F G H I K L M,
Bet A C1 C -> Defect A B C1 D E F -> Defect B C1 C G H I ->
SumA D E F G H I K L M ->
SAMS D E F G H I /\ Defect A B C K L M.