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 filterdiff_mult_fct {K : AbsRing} {U : NormedModule K} (f g : U -> K) x (lf lg : U -> K) : (forall (n m : K), mult n m = mult m n) -> filterdiff f (locally x) lf -> filterdiff g (locally x) lg -> filterdiff (fun t => mult (f t) (g t)) (locally x) (fun t => plus (mult (lf t) (g x)) (mult (f x) (lg t))).