simple induction n.
simpl in |- *.
red in |- *.
intros.
inversion x.
inversion in_range_prf.
intros.
simpl in |- *.
red in |- *.
intro x; elim x; intro i; case i.
simpl in |- *.
intro.
apply Zero_times_a_vector_gives_zero; auto with algebra.
simpl in |- *.
intros.
apply Zero_times_a_vector_gives_zero; auto with algebra.
Qed.

Lemma mult_const_zero :
forall (n : Nat) (a : seq n F),
mult_by_scalars a (const_seq n (zero V)) =' const_seq n (zero V) in seq _ _.