Require Export GeoCoq.Meta_theory.Parallel_postulates.SPP_ID.
Require Export GeoCoq.Meta_theory.Parallel_postulates.TCP_tarski.
Require Export GeoCoq.Meta_theory.Parallel_postulates.alternate_interior_angles_proclus.
Require Export GeoCoq.Meta_theory.Continuity.angle_archimedes.
Require Export GeoCoq.Meta_theory.Continuity.archimedes.
Require Export GeoCoq.Meta_theory.Continuity.aristotle.
Require Export GeoCoq.Meta_theory.Parallel_postulates.euclid_5_original_euclid.
Require Export GeoCoq.Meta_theory.Parallel_postulates.existential_triangle_rah.
Require Export GeoCoq.Meta_theory.Parallel_postulates.par_perp_perp_TCP.
Require Export GeoCoq.Meta_theory.Parallel_postulates.playfair_alternate_interior_angles.
Require Export GeoCoq.Meta_theory.Parallel_postulates.playfair_bis_playfair.
Require Export GeoCoq.Meta_theory.Parallel_postulates.playfair_universal_posidonius_postulate.
Require Export GeoCoq.Meta_theory.Parallel_postulates.proclus_SPP.
Require Export GeoCoq.Meta_theory.Parallel_postulates.rah_triangle.
Require Export GeoCoq.Meta_theory.Parallel_postulates.tarski_euclid.
Require Export GeoCoq.Meta_theory.Parallel_postulates.triangle_playfair_bis.
Require Export GeoCoq.Meta_theory.Parallel_postulates.triangle_existential_triangle.
Require Export GeoCoq.Meta_theory.Parallel_postulates.universal_posidonius_postulate_par_perp_perp.
Require Export GeoCoq.Tarski_dev.Annexes.defect.
Section Legendre.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.

Lemma legendre_aux2 :
~ hypothesis_of_obtuse_saccheri_quadrilaterals ->
forall A B C,
~ Col A B C ->  Acute B A C ->
(forall T,
InAngle T B A C ->
exists X Y : Tpoint, Out A B X /\ Out A C Y /\ Bet X T Y) ->
forall P Q R S T U,
Defect A B C P Q R -> GradAExp P Q R S T U ->
exists B' C' P' Q' R',
(Out A B B' /\ Out A C C' /\
Defect A B' C' P' Q' R' /\ LeA S T U P' Q' R').