End NormedModule1.
Section NormedModule2.
Context {T : Type} {K : AbsRing} {V : NormedModule K}.

Lemma filterlimi_locally_unique :
forall {F} {FF : ProperFilter' F} (f : T -> V -> Prop) (x y : V),
F (fun x => forall y1 y2, f x y1 -> f x y2 -> y1 = y2) ->
filterlimi f F (locally x) -> filterlimi f F (locally y) ->
x = y.