Section prod_ModuleSpace.
Context {K : Ring} {U V : ModuleSpace K}.
Definition prod_scal (x : K) (u : U * V) :=
(scal x (fst u), scal x (snd u)).

Lemma prod_scal_distr_r :
forall (x y : K) (u : U * V),
prod_scal (plus x y) u = prod_plus (prod_scal x u) (prod_scal y u).