(** * 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 Derive_partial_derive_aux1: forall p f x y, locally_2d (ex_diff_n f (S p)) x y -> partial_derive 1 p f x y = partial_derive 0 p (partial_derive 1 0 f) x y.