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_l :
forall (x : K) (u v : U * V),
prod_scal x (prod_plus u v) = prod_plus (prod_scal x u) (prod_scal x v).