intros; simpl in |- *.
red in |- *; simpl in |- *.
intros.
apply lin_comb_casting with W; auto with algebra.
Qed.

Lemma lin_comb_scalar_multiplied :
forall (F : field) (V : vectorspace F) (x : V) (n : Nat)
(a : seq n F) (v : seq n V) (c : F),
x =' c mX sum (mult_by_scalars a v) in _ -> is_lin_comb x (seq_set v).