unfold colineaire 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 zero (plus (double (cons u v)) (double (cons v v'))));
auto.
apply compatible; auto.
cut (colineaire u' u); intros; auto.
apply transitive with (plus (double (cons u v)) (double (cons v v'))); 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 colineaire_modulo_pi.

Lemma orthogonal_opp : forall u v : V, orthogonal u v -> orthogonal u (opp v).
unfold orthogonal in |- *; intros.