(** * Schwarz theorem *)

Lemma Schwarz :
forall (f : R -> R -> R) x y,
locally_2d (fun u v =>
ex_derive (fun z : R => f z v) u /\
ex_derive (fun z : R => f u z) v /\
ex_derive (fun z : R => Derive (fun t => f z t) v) u /\
ex_derive (fun z : R => Derive (fun t => f t z) u) v) x y ->
continuity_2d_pt (fun u v => Derive (fun z => Derive (fun t => f z t) v) u) x y ->
continuity_2d_pt (fun u v => Derive (fun z => Derive (fun t => f t z) u) v) x y ->
Derive (fun z => Derive (fun t => f z t) y) x = Derive (fun z => Derive (fun t => f t z) x) y.