End Operations_fct.

(* Specifications: *)
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))).
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).