End METRIC_BASICS.
Section METRIC_PROPERTIES.

(* Specifications: *)
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.