induction n.
intros.
apply False_ind; auto with algebra.
intros.
case i.
destruct index as [| n0].
simpl in |- *.
auto with algebra.
intro.
apply
Trans
with
(modify_seq (Seqtl v) (Build_finiteT (lt_S_n _ _ in_range_prf)) a
(Build_finiteT (lt_S_n _ _ in_range_prf)));
auto with algebra.
Qed.
Hint Resolve modify_seq_defprop: algebra.

Lemma modify_seq_modifies_one_elt :
forall (A : Setoid) (n : Nat) (v : seq n A) (i : fin n) (a : A) (j : fin n),
~ j =' i in _ -> modify_seq v i a j =' v j in _.