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.

Lemma ex_filterdiff_mult_fct {K : AbsRing} {U : NormedModule K} (f g : U -> K) x : (forall (n m : K), mult n m = mult m n) -> ex_filterdiff f (locally x) -> ex_filterdiff g (locally x) -> ex_filterdiff (fun t => mult (f t) (g t)) (locally x).