intros a x; try assumption.
apply Sym.
apply GROUP_law_inverse.
apply Trans with (module_mult (sgroup_law R a (group_inverse R a)) x);
auto with algebra.
apply Trans with (module_mult (monoid_unit R) x); auto with algebra.
apply MODULE_absorbant_l.
Qed.
Variable Mod' : MODULE R.
Variable f : Hom Mod Mod'.

Lemma MODULE_hom_prop :
forall (a : R) (x : Mod), Equal (f (module_mult a x)) (module_mult a (f x)).