End Filterlim_switch.

(* Specifications: *)
Lemma filterlim_switch_dom {T1 T2 : Type} {U : CompleteSpace}
F1 (FF1 : ProperFilter F1) F2 (FF2 : Filter F2)
(dom : T2 -> Prop) (HF2 : forall P, F2 P -> exists x, dom x /\ P x)
(f : T1 -> T2 -> U) g h :
filterlim (fun x (y : {z : T2 | dom z}) => f x (proj1_sig y)) F1 (locally (T := fct_UniformSpace _ _) (fun y : {z : T2 | dom z} => g (proj1_sig y))) ->
(forall x, filterlim (f x) (within dom F2) (locally (h x))) ->
exists l : U, filterlim h F1 (locally l) /\ filterlim g (within dom F2) (locally l).