Definition rel_base (b : pred T) :=
  forall x' y', ~~ b (h x') -> e (h x') (h y') = e' x' y'.

Lemma map_path b x' p' (Bb : rel_base b) : ~~ has (preim h b) (belast x' p') -> path e (h x') (map h p') = path e' x' p'.