proposition: Lemma is_derive_n_sum_n n (f : nat -> R -> R) (k: nat) (x : R) : locally x (fun t => forall l j , (l <= n)%nat ->(j <= k)%nat -> ex_derive_n (f l) j t) -> is_derive_n (fun y => sum_n (fun j => f j y) n) k x (sum_n (fun j => Derive_n (f j) k x) n).
proof: Proof. intros. apply is_derive_n_sum_n_m. move: H ; apply filter_imp ; intros. apply H => //. by apply H0. Qed.
imports: Coq(Reals Psatz ssreflect) Rcomplements Rbar Lim_seq Iter Hierarchy Continuity Equiv