Section prod_NormedModule.
Context {K : AbsRing} {U V : NormedModule K}.
Definition prod_norm (x : U * V) :=
sqrt (norm (fst x) ^ 2 + norm (snd x) ^ 2).
Definition prod_norm_factor :=
sqrt 2 * Rmax (@norm_factor K U) (@norm_factor K V).

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