(** * Topology on [R]² *)
Definition locally_2d (P : R -> R -> Prop) x y :=
exists delta : posreal, forall u v, Rabs (u - x) < delta -> Rabs (v - y) < delta -> P u v.

Lemma locally_2d_ex_dec :
forall P x y,
(forall x y, P x y \/ ~P x y) ->
locally_2d P x y ->
{d : posreal | forall u v, Rabs (u-x) < d -> Rabs (v-y) < d -> P u v}.