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_scal : forall {F : (K * V -> Prop) -> Prop} {FF : ProperFilter F} (x : K * V), is_filter_lim F x -> (forall (n m : K), mult n m = mult m n) -> filterdiff (fun t : K * V => scal (fst t) (snd t)) F (fun t => plus (scal (fst t) (snd x)) (scal (fst x) (snd t))).