Definition pos_div_2 (eps : posreal) := mkposreal _ (is_pos_div_2 eps).(** * The sign function *)

Definition sign (x : R) :=
  match total_order_T 0 x with
  | inleft (left _) => 1
  | inleft (right _) => 0
  | inright _ => -1
  end.

Lemma Riemann_integrable_plus : forall (f g : R -> R) (a b : R), Riemann_integrable f a b -> Riemann_integrable g a b -> Riemann_integrable (fun x => f x + g x) a b.