{A B} (f : A -> B) (C : B -> Type) : (forall b, ooExtendableAlong (const_tt (hfiber f b)) (fun _ => C b)) -> ooExtendableAlong f C. Proof. intros orth n; revert C orth. induction n as [|n IHn]; intros C orth; [exact tt | split]. - intros g. exists (fun b => (fst (orth b 1%nat) (fun x => x.2 # g x.1)).1 tt). intros a. rewrite (path_unit tt (const_tt _ a)). exact ((fst (orth (f a) 1%nat) _).2 (a ; 1)). - intros h k. apply IHn; intros b. apply ooextendable_homotopy, orth. Defined.
Require Import HoTT.Basics HoTT.Types. Require Import Equiv.PathSplit Homotopy.IdentitySystems. Require Import Cubical.DPath Cubical.DPathSquare. Require Import Colimits.Coeq Colimits.MappingCylinder.