Definition subitv (i1 i2 : interval R) :=
  match i1, i2 with
    | Interval a1 b1, Interval a2 b2 => le_boundl a2 a1 && le_boundr b1 b2
  end.

Lemma subitvPr (a b1 b2 : itv_bound R) : le_boundr b1 b2 -> {subset (Interval a b1) <= (Interval a b2)}.