intro.
elim P.
intros.
unfold pred_compatible in Pred_compatible_prf.
apply Pred_compatible_prf with (hdtl w); auto with algebra.
unfold hdtl in |- *.
apply H.
Qed.

(* Specifications: *)
Lemma cons_lemma_verynice :
forall (P : Predicate (seq (S n) A)) (H : 0 < S n) (w : seq (S n) A),
Pred_fun P (w (Build_finiteT H);; Seqtl w) -> Pred_fun P w.