unfold orthogonal in |- *; intros.
apply
transitive with (double (plus (cons u u') (plus (cons u' v') (cons v' v))));
auto.
apply
transitive
with (plus (double (cons u u')) (double (plus (cons u' v') (cons v' v))));
auto.
apply
transitive
with
(plus (double (cons u u'))
(plus (double (cons u' v')) (double (cons v' v))));
auto.
apply compatible; auto.
apply transitive with (plus pi (plus (double (cons u' v')) pi)); auto.
apply compatible; auto.
apply compatible; auto.
apply regulier with (a := double (cons v v')) (c := pi); auto.
apply transitive with (double (plus (cons v v') (cons v' v))); auto.
apply transitive with (double (cons v v)); auto.
apply transitive with (double zero); auto.
apply transitive with zero; auto.
apply transitive with (plus (plus (double (cons u' v')) pi) pi); auto.
apply transitive with (plus (double (cons u' v')) (plus pi pi)); auto.
apply transitive with (plus (double (cons u' v')) zero); auto.
apply compatible; auto.
apply transitive with (plus zero (double (cons u' v'))); auto.
Qed.
Hint Resolve double_orthogonal.

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