Definition image_chain := Chain image_chainP.
End ImageChain.
Notation "[ f '^^' s 'by' M ]" := (@image_chain _ _ s f M)
(at level 0, format "[ f  '^^'  s  'by'  M ]") : form_scope.
(* some chain constructions *)
Section ChainId.
Variables (T : poset) (s : chain T).

Lemma id_chainE (M : monotone id) : [id ^^ s by M] = s.
End ChainConst.