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

(** In a convex quadrilateral ABCD, the sum of the defects of ABC and ADC is equal to the sum of the defects of ABD and CBD. We add some hypotheses to make the proof easier *)
Lemma t22_16_2 :
~ hypothesis_of_obtuse_saccheri_quadrilaterals ->
forall A B C D A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3 O P Q R,
Defect A B C D1 D2 D3 -> Defect A B D C1 C2 C3 ->
Defect A D C B1 B2 B3 -> Defect C B D A1 A2 A3 ->
Bet A O C -> Bet B O D ->
SAMS D1 D2 D3 B1 B2 B3 -> SumA D1 D2 D3 B1 B2 B3 P Q R ->
SAMS C1 C2 C3 A1 A2 A3 /\ SumA C1 C2 C3 A1 A2 A3 P Q R.