intros.
red in |- *.
exists n.
set (axc_fun := fun i : fin n => c rX a i) in *.
assert (axc_comp : fun_compatible axc_fun).
red in |- *.
intros.
unfold axc_fun in |- *.
apply RING_comp; auto with algebra.
set (axc := Build_Map axc_comp:seq _ _) in *.
exists axc.
exists (seq_set_seq v).
apply Trans with (sum (mult_by_scalars axc v)).
apply Trans with (c mX sum (mult_by_scalars a v)); auto with algebra.
apply
Trans with (sum (mult_by_scalars (const_seq n c) (mult_by_scalars a v)));
auto with algebra.
apply sum_comp; auto with algebra.
simpl in |- *.
red in |- *; intros.
unfold mult_by_scalars, axc in |- *.
simpl in |- *.
unfold axc_fun in |- *.
auto with algebra.
apply sum_comp; auto with algebra.
Qed.

(* Specifications: *)
Lemma lin_comb_omit :
forall (F : field) (V : vectorspace F) (x : V) (n : Nat)
(a : seq n F) (v : seq n V) (i : fin n),
x =' sum (omit (mult_by_scalars a v) i) in _ -> is_lin_comb x (seq_set v).