Section Diff.
Context {K : AbsRing} {U : NormedModule K} {V : NormedModule K}.
Definition filterdiff (f : U -> V) F (l : U -> V) :=
is_linear l /\ forall x, is_filter_lim F x ->
is_domin F (fun y : U => minus y x) (fun y => minus (minus (f y) (f x)) (l (minus y x))).
Definition ex_filterdiff (f : U -> V) F :=
exists (l : U -> V), filterdiff f F l.
(** ** Operations *)

Lemma ex_filterdiff_ext_locally (f g : U -> V) x :
locally x (fun y => f y = g y)
-> ex_filterdiff f (locally x) -> ex_filterdiff g (locally x).