(** * Iterated differential *)

Fixpoint ex_diff_n f n x y :=
  continuity_2d_pt f x y /\
  match n with
  | O => True
  | S n =>
    ex_derive (fun z => f z y) x /\
    ex_derive (fun z => f x z) y /\
    ex_diff_n (fun u v => Derive (fun z => f z v) u) n x y /\
    ex_diff_n (fun u v => Derive (fun z => f u z) v) n x y
  end.

Lemma ex_diff_n_continuity_inf_2 : forall n p k, (p+k < n)%nat -> forall f x y, ex_diff_n f n x y -> continuity_2d_pt (fun u v => Derive (fun z : R => partial_derive p k f u z) v) x y.