(** * Definitions on [R] *)

Definition Derive (f : R -> R) (x : R) := real (Lim (fun h => (f (x+h) - f x)/h) 0).

Lemma ex_derive_sum_n : forall (f : nat -> K -> V) (n : nat) (x : K), (forall k, (k <= n)%nat -> ex_derive (f k) x) -> ex_derive (fun y => sum_n (fun k => f k y) n) x.