unfold colineaire, orthogonal in |- *; intros.
apply transitive with (double (plus (cons u v) (cons v w))); auto.
apply transitive with (plus (double (cons u v)) (double (cons v w))); auto.
apply transitive with (plus pi zero); auto.
apply compatible; auto.
apply transitive with (plus zero pi); auto.
Qed.

Lemma colineaire_transitive :
forall u v w : V, colineaire u v -> colineaire v w -> colineaire u w.