(** Addition of functions *)

Lemma is_derive_n_sum_n_m n m (f : nat -> R -> R) (k: nat) (x : R) :
locally x (fun t => forall l j , (n <= l <= m)%nat ->(j <= k)%nat -> ex_derive_n (f l) j t) ->
is_derive_n (fun y => sum_n_m (fun j => f j y) n m) k x
(sum_n_m (fun j => Derive_n (f j) k x) n m).