intros u v w t H; try assumption.
apply transitive with (plus zero (cons v u)).
apply symetrique; auto.
apply zero_plus.
apply transitive with (plus (plus (cons v u) (cons u v)) (cons v u)).
apply compatible.
apply symetrique; auto.
apply permute.
apply reflexive.
apply transitive with (plus (cons v u) (plus (cons w t) (cons t w))).
apply transitive with (plus (cons v u) (plus (cons u v) (cons v u))).
apply symetrique; auto.
apply point_angle.plus_assoc.
apply compatible.
apply reflexive.
apply transitive with zero.
apply permute.
apply symetrique; auto.
apply permute.
apply transitive with (plus (plus (cons v u) (cons w t)) (cons t w)).
apply point_angle.plus_assoc.
apply transitive with (plus zero (cons t w)).
apply compatible.
apply transitive with (plus (cons v u) (cons u v)).
apply compatible.
apply reflexive.
apply symetrique; auto.
apply permute.
apply reflexive.
apply zero_plus.
Qed.

Lemma regulier_cons :
forall u v w t s : V,
R (cons u v) (cons u w) ->
R (cons u t) (cons u s) -> R (cons v t) (cons w s).