intros.
apply concat_prop_per_part.
intros.
generalize (H i Hi); intro.
apply
(Pred_compatible_prf (p:=P))
with ((v ++ w) (Build_finiteT (lt_plus_trans _ _ m Hi))).
apply H1.
apply Sym.
apply concat_first_part.
intros.
cut (n + j < n + m).
intro Hj'.
apply (Pred_compatible_prf (p:=P)) with ((v ++ w) (Build_finiteT Hj'));
auto with algebra.
auto with arith.
Qed.

(* Specifications: *)
Lemma split_to_concat :
forall vw : seq (n + m) A, sigT (fun a => sigT (fun b => vw =' a ++ b in _)).