Qed.

Lemma level_remake_rmap: forall f g lev H Hg, @level rmap _ (proj1_sig (remake_rmap f g lev H Hg)) = lev.
refine (fun f g lev H Hg => match proj2_sig (remake_rmap f g lev H Hg) with
| conj LEVEL _ => LEVEL
end).