(O O' : ReflectiveSubuniverse) `{O' <= Sep O} {A B C A' B' C'} (f : A -> B) (g : A -> C) {f' : A' -> B'} {g' : A' -> C'} (h : A -> A') (k : B -> B') (l : C -> C') (p : k o f == f' o h) (q : l o g == g' o h) `{IsConnMap O _ _ h} `{IsConnMap O' _ _ k} `{IsConnMap O' _ _ l} : IsConnMap O' (functor_pushout h k l p q). Proof. apply conn_map_from_extension_elim; intros. apply extension_functor_coeq. - apply extendable_functor_sum; rapply ooextendable_conn_map_inO. - pose (inO_leq O' (Sep O)); intros; rapply ooextendable_conn_map_inO. 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.