apply transitive with (double (plus (cons u v) (cons v (opp v)))); auto.
apply transitive with (plus (double (cons u v)) (double (cons v (opp v))));
auto.
apply transitive with (plus pi (double pi)); auto.
apply compatible; auto.
apply transitive with (plus pi zero); auto.
apply compatible; auto.
apply transitive with (plus zero pi); auto.
Qed.
Hint Resolve orthogonal_opp.

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