apply regulier with (a := double (cons u v)) (c := double (cons u v)); auto.
apply transitive with (double (plus (cons u v) (cons v u))); auto.
apply transitive with (double zero); auto.
apply transitive with (plus zero zero); auto.
apply compatible; auto.
Qed.
Hint Resolve colineaire_sym.

Lemma colineaire_modulo_pi :
forall u v u' v' : V,
colineaire u u' ->
colineaire v v' -> R (double (cons u' v')) (double (cons u v)).