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}.

Theorem legendre_s_first_theorem :
archimedes_axiom ->
forall A B C D E F,
SumA C A B A B C D E F ->
SAMS D E F B C A.