{A B : Type} (f : A -> B) `{IsConnMap O _ _ f} (P : B -> Type) `{forall b:B, In O (P b)} (d : forall a:A, P (f a)) : forall b:B, P b. Proof. intros b. refine (pr1 (isconnected_elim O (A:=hfiber f b) _ _)). intros [a p]. exact (transport P p (d a)). Defined.
Require Import HoTT.Basics HoTT.Types. Require Import Equiv.BiInv Extensions HProp HFiber NullHomotopy Limits.Pullback. Require Import Homotopy.IdentitySystems. Require Import Colimits.Pushout.