(** * 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_impl_strong :
forall (P Q : R -> R -> Prop) x y, locally_2d (fun u v => locally_2d P u v -> Q u v) x y ->
locally_2d P x y -> locally_2d Q x y.