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_sym :
forall (x y : U * V) (eps : R),
prod_ball x eps y -> prod_ball y eps x.