Definition prod_norm_factor :=
sqrt 2 * Rmax (@norm_factor K U) (@norm_factor K V).

(* Specifications: *)
Lemma prod_norm_compat2 :
forall (x y : U * V) (eps : posreal),
ball x eps y ->
prod_norm (minus y x) < prod_norm_factor * eps.
Lemma prod_norm_eq_zero :
forall x : U * V,
prod_norm x = 0 -> x = zero.