intros.
unfold mult_by_scalars in |- *.
apply (pointwise_comp_general (A:=fin N) (B:=R) (C:=V) (D:=V));
auto with algebra.
Qed.

Lemma mult_by_scalars_fun2_compatible :
forall (R : ring) (V : module R) (N : Nat),
fun2_compatible
(mult_by_scalars (V:=V) (N:=N):MAP _ _ -> MAP _ _ -> MAP _ _).