Lemma legendre_aux : ~ hypothesis_of_obtuse_saccheri_quadrilaterals -> forall A B C D B1 C1 P Q R S T U V W X, ~ Col A B C -> CongA A C B C B D -> Cong A C B D -> TS B C A D -> Out A B B1 -> Out A C C1 -> Bet B1 D C1 -> Defect A B C P Q R -> Defect A B1 C1 S T U -> SumA P Q R P Q R V W X -> SAMS P Q R P Q R /\ LeA V W X S T U.