Require Export C15_Parallelogramm.
Section DISTANCE_PROPERTIES.

(* Specifications: *)
Lemma LSltNotRefl : forall A B : Point,
~LSlt (Distance A B) (Distance A B).
Lemma LSltNotEq : forall A B C D,
LSlt (Distance A B) (Distance C D) ->
(Distance A B) <> (Distance C D).
Lemma LSplusComm : forall A B C D : Point,
A <> B ->
C <> D ->
LSplus (Distance A B) (Distance C D) = LSplus (Distance C D) (Distance A B).
Lemma LSplusAss : forall A B C D E F : Point,
A <> B ->
C <> D ->
E <> F ->
LSplus (Distance A B) (LSplus (Distance C D) (Distance E F)) =
LSplus(LSplus (Distance A B)  (Distance C D)) (Distance E F).
Lemma LSltLSplus : forall A B C D : Point,
LSlt LS0 (Distance A B) ->
LSlt LS0 (Distance C D) ->
LSlt LS0 (LSplus (Distance A B) (Distance C D)).
Lemma ChaslesLSltTrans : forall A B C D : Point,
A <> B ->
HalfLine A B C ->
HalfLine A C D ->
LSlt (Distance A B) (Distance A C) ->
LSlt (Distance A C) (Distance A D) ->
LSlt (Distance A B) (Distance A D).
Lemma LSltTrans : forall A B C D E F : Point,
A <> B ->
C <> D ->
E <> F ->
LSlt (Distance A B) (Distance C D) ->
LSlt (Distance C D) (Distance E F) ->
LSlt (Distance A B) (Distance E F).