(** * 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_1d_strong :
forall (P : R -> R -> Prop) x y,
locally_2d P x y ->
locally_2d (fun u v => forall t, 0 <= t <= 1 ->
locally t (fun z => locally_2d P (x + z * (u - x)) (y + z * (v - y)))) x y.