Definition LS0 := Distance Oo Oo.

(* Specifications: *)
Lemma NullDist : forall A, Distance A A = LS0.
Lemma NullLSlt : forall A B : Point, A <> B -> LSlt LS0 (Distance A B).
Lemma EquiOrientedNotEquiOriented : forall A B C D : Point, A <> B -> EquiOriented A B C D -> ~EquiOriented A B D C.
Lemma LSltDistinct : forall A B C D, LSlt (Distance A B) (Distance C D) -> C <> D.
Lemma LSltNull : forall A B, LSlt LS0 (Distance A B) -> A <> B.
Lemma NotLSltNullNull : ~LSlt LS0 LS0.
Lemma LS0NeutralRight : forall A B : Point, LSplus (Distance A B) LS0 = Distance A B.
Lemma LS0NeutralLeft : forall A B : Point, LSplus LS0 (Distance A B) = Distance A B.
Lemma EquiOrientedNotEquiOrientedABC : forall A B C : Point, A <> B -> EquiOriented A B B C -> ~EquiOriented A C C B.
Lemma HalfLineAntisymLSlt : forall A B C : Point, A <> B -> HalfLine A B C -> LSlt (Distance A B) (Distance A C) -> ~LSlt (Distance A C) (Distance A B).
Lemma AntisymLSlt : forall A B C : Point, A <> B -> LSlt (Distance A B) (Distance A C) -> ~LSlt (Distance A C) (Distance A B).
Lemma ClockwiseNotNullSide : forall A B C : Point, Clockwise A B C -> ~Distance B C = LS0.
Lemma DistDistinctNull : forall A B D : Point, Distance A B = LS0 -> D <> A -> A = B.
Lemma DistNull : forall A B, Distance A B = LS0 -> A = B.
Lemma EquiDistantDistinct : forall A B C D : Point, A <> B -> Distance A B = Distance C D -> C <> D.
Lemma BetweenLSlt : forall A B C, (Between A B C) -> LSlt (Distance A B) (Distance A C).
Lemma LSltBetween : forall A B C, A <> B -> HalfLine A B C -> LSlt (Distance A B) (Distance A C) -> (Between A B C).
Lemma DistDistinct : forall A B, Distance A B <> LS0 -> A <> B.
Lemma DistinctDist : forall A B, A <> B -> Distance A B <> LS0.
Lemma ChaslesComm : forall A B C : Point, HalfLine A B C -> HalfLine C B A -> LSplus (Distance A B) (Distance B C) = LSplus (Distance B C) (Distance A B).
Lemma ChaslesAssoc : forall A B C D : Point, HalfLine A B C -> HalfLine C B A -> HalfLine A B D -> HalfLine D B A -> HalfLine B C D -> HalfLine D C B -> HalfLine A C D -> HalfLine D C A -> LSplus (Distance A B) (LSplus (Distance B C) (Distance C D)) = LSplus (LSplus (Distance A B) (Distance B C)) (Distance C D).
Lemma SSSEqualBD : forall A B C D : Point, Clockwise A B C -> Clockwise A D C -> Distance A B = Distance A D -> Distance B C = Distance D C -> B = D.
Lemma SSSEqualCD : forall A B C D : Point, Clockwise A B C -> Clockwise A B D -> Distance A C = Distance A D -> Distance B C = Distance B D -> C = D.