intros u v H; unfold orthogonal, double in |- *.
apply transitive with (plus (cons u v) (cons v (opp u))); auto.
apply compatible; auto.
apply transitive with (cons u (opp u)); auto.
Qed.

(* Specifications: *)
Lemma critere_orthogonal_reciproque :
forall u v : V, orthogonal u v -> R (cons u v) (cons v (opp u)).