Definition Supplementary (alpha beta : AS) :=
exists A  : Point,  exists B  : Point,  exists C : Point,  exists D : Point, 
	A <> C /\ Between B A D /\ Angle B A C = alpha /\ Angle C A D = beta.

(* Specifications: *)
Lemma SupplementaryCommut : forall alpha beta : AS, Supplementary alpha beta -> Supplementary beta alpha.
Lemma HalfLineBetweenBetween : forall A B C D : Point, Between A C D -> HalfLine C A B -> Between B C D.
Lemma SupplementBC : forall A B C D A' B' C' D' : Point, A <> C -> A' <> C' -> B <> C -> Between B A D -> Between B' A' D' -> Angle B A C = Angle B' A' C' -> Angle C A D = Angle C' A' D'.
Lemma Supplement : forall A B C D A' B' C' D' : Point, A <> C -> A' <> C' -> Between B A D -> Between B' A' D' -> Angle B A C = Angle B' A' C' -> Angle C A D = Angle C' A' D'.
Lemma SupplementaryEq : forall alpha beta gamma : AS, Supplementary alpha beta -> Supplementary alpha gamma -> beta = gamma.
Lemma OpposedAtVertex : forall A B C D E : Point, Between B A D -> Between C A E -> Angle B A C = Angle D A E.
Lemma CongruentOpposedTriangles : forall A B C D I : Point, Between A I C -> Between B I D -> Distance I A = Distance I C -> Distance I B = Distance I D -> CongruentTriangles A I B C I D.
Lemma SupplementaryRec : forall A B C D : Point, Clockwise B A C -> Clockwise C A D -> Supplementary (Angle B A C) (Angle C A D) -> Between B A D.