Section SubCPO.
Variables (D : cpo) (s : Pred D) (C : chain_closed s).
Local Notation tp := (subPoset s).
Definition sub_bot := exist _ (@bot D) (proj1 C).

(* applied lim equals the lim of applications *)
(* ie., part of continuity of application *)
(* but is so often used, I give it a name *)
Lemma lim_appE A (D : cpo) (s : chain [cpo of A -> D]) (x : A) :
lim s x = lim (app_chain s x).