(** * Schwarz theorem *)

Lemma Schwarz_aux :
forall f x y (eps : posreal),
( forall u v, Rabs (u - x) < eps -> Rabs (v - y) < eps ->
ex_derive (fun z : R => f z v) u /\
ex_derive (fun z : R => Derive (fun t => f t z) u) v ) ->
forall h k, Rabs h < eps -> Rabs k < eps ->
let phi k x := f x (y + k) - f x y in
exists u, exists v,
Rabs (u - x) <= Rabs h /\ Rabs (v - y) <= Rabs k /\
phi k (x + h) - phi k x = h * k * (Derive (fun z => Derive (fun t => f t z) u) v).