intros a b c d; try assumption.
apply transitive with (plus a (plus (plus b c) d)).
apply symetrique.
apply point_angle.plus_assoc.
apply transitive with (plus (plus a (plus b c)) d).
apply point_angle.plus_assoc.
apply transitive with (plus (plus (plus a b) c) d).
apply compatible.
apply point_angle.plus_assoc.
apply reflexive.
apply transitive with (plus (plus a b) (plus c d)).
apply symetrique.
apply point_angle.plus_assoc.
apply transitive with (plus (plus a b) (plus d c)).
apply compatible.
apply reflexive.
apply plus_sym.
apply transitive with (plus (plus (plus a b) d) c).
apply transitive with (plus (plus a b) (plus d c)).
apply reflexive.
apply point_angle.plus_assoc.
apply compatible.
apply symetrique.
apply point_angle.plus_assoc.
apply reflexive.
Qed.

Lemma somme_pi :
forall u v w : V,
R (plus (cons u v) (plus (cons w (opp u)) (cons (opp v) (opp w)))) pi.