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_fct x (f : U -> K) (g : U -> V) lf lg : (forall (n m : K), mult n m = mult m n) -> filterdiff f (locally x) lf -> filterdiff g (locally x) lg -> filterdiff (fun t => scal (f t) (g t)) (locally x) (fun t => plus (scal (lf t) (g x)) (scal (f x) (lg t))).