intros.
intro i.
destruct i.
destruct index as [| n0].
simpl in |- *.
auto with algebra.
rename in_range_prf into p.
apply Trans with ((v ++ v') (Build_finiteT (lt_S_n _ _ p)));
auto with algebra.
apply Trans with ((Seqtl (a;; v) ++ v') (Build_finiteT (lt_S_n _ _ p)));
auto with algebra.
Qed.

(* Specifications: *)
Lemma concat_first_element :
forall (v : seq (S n) A) (w : seq m A) (Hnm : 0 < S (n + m)) (Hn : 0 < S n),
(v ++ w) (Build_finiteT Hnm) =' v (Build_finiteT Hn) in _.