intros a b c; try assumption.
elim non_vide_V; intros u H; clear H.
elim angle_cons with (a := a) (u := u); intros v H; try exact H.
elim angle_cons with (a := b) (u := v); intros v0 H0; try exact H0.
elim angle_cons with (a := c) (u := v0); intros v1 H1; try exact H1.
apply transitive with (plus (cons u v) (plus (cons v v0) (cons v0 v1))).
apply compatible; auto.
apply compatible; auto.
apply transitive with (cons u v1).
apply symetrique; auto.
apply Chasles_2.
apply transitive with (plus (cons u v0) (cons v0 v1)).
apply symetrique; auto.
apply Chasles.
apply compatible; auto.
apply transitive with (plus (cons u v) (cons v v0)).
apply symetrique; auto.
apply Chasles.
apply compatible; auto.
apply symetrique; auto.
apply symetrique; auto.
apply symetrique; auto.
Qed.

Lemma compatible_croix :
forall a b c d : AV, R a b -> R c d -> R (plus a d) (plus b c).