Section prod_UniformSpace.
Context {U V : UniformSpace}.
Definition prod_ball (x : U * V) (eps : R) (y : U * V) :=
ball (fst x) eps (fst y) /\ ball (snd x) eps (snd y).

Lemma prod_ball_center :
forall (x : U * V) (eps : posreal),
prod_ball x eps x.