intros.
induction H.
apply False_ind; auto with algebra.
case (IHis_subseq i).
intros.
destruct x.
exists (Build_finiteT (lt_n_S _ _ in_range_prf)).
apply Trans with (v (Build_finiteT in_range_prf)); auto with algebra.
destruct i; destruct index as [| n0].
exists (Build_finiteT (lt_O_Sn n)).
auto with algebra.
generalize (IHis_subseq (Build_finiteT (lt_S_n _ _ in_range_prf))); intro.
inversion_clear H0.
destruct x.
exists (Build_finiteT (lt_n_S _ _ in_range_prf0)).
apply Trans with (v (Build_finiteT in_range_prf0)); auto with algebra.
generalize (IHis_subseq i).
intros.
inversion_clear H2.
exists x.
apply Trans with (w' i); auto with algebra.
apply Trans with (v' x); auto with algebra.
Qed.

Lemma subseqs_are_not_longer :
forall (m n : Nat) (w : seq m A) (v : seq n A), is_subseq w v -> m <= n.