intros. apply seq_set_head_or_tail.
2:exact(P4helpr n G A AH AH' AH'' c V Va Vb Vc V' V'a i_ i_H j g gH).
apply in_part_comp_r with A. 2:exact(P4helprr n G A AH AH' AH'').
elim(classic(i_ =' j)).
intro H.
assert(H1:in_part g(minus_part(V i_)(subtype_elt(head AH)))).
apply in_part_comp_r with
(modify_seq V i_(minus_part(V i_)(subtype_elt(head AH)))j).
exact gH. apply Trans with
(modify_seq V i_(minus_part(V i_)(subtype_elt(head AH)))i_).
apply Map_compatible_prf. apply Sym. exact H.
apply modify_seq_defprop. simpl in H1. elim H1. intros H1a H1b.
apply (Va i_). exact H1a.
intro H.
assert(H1:in_part g(V j)). apply in_part_comp_r with
(modify_seq V i_(minus_part(V i_)(subtype_elt(head AH)))j).
exact gH. apply modify_seq_modifies_one_elt. unfold not.
intro H'. apply H. apply Sym. exact H'. apply(Va j). exact H1.
Qed.

(* Specifications: *)
Lemma seqtlobv:forall(n:Nat)(G:Setoid)(A:part_set G)
(AH:seq(S n)A)
(g:G)(gH:in_part g (seq_set(Seqtl(Map_embed AH)))),
in_part g A.