intros.
unfold mult_by_scalars in |- *.
apply Sym; auto with algebra.
Qed.
Hint Resolve mult_by_scalars_Seqtl: algebra.

Lemma mult_by_scalars_concat :
forall (R : ring) (V : module R) (N M : Nat) (a : Map (fin N) R)
(b : Map (fin M) R) (v : Map (fin N) V) (w : Map (fin M) V),
mult_by_scalars (a ++ b) (v ++ w) ='
mult_by_scalars a v ++ mult_by_scalars b w in _.