intros. elim(Vb g(seqtlobv n G A AH g gH)). intros j jH.
elim(classic(i_ =' j)). intro H. exists i_.
apply in_part_comp_r with(minus_part(V i_)(subtype_elt(head AH))).
2:apply Sym. 2:apply modify_seq_defprop.
simpl. split. apply in_part_comp_r with(V j). exact jH.
apply Map_compatible_prf. apply Sym. exact H.
exact (seq_set_head_or_tail_inv n G A AH AH' AH'' g gH).
intro H. exists j. apply in_part_comp_r with (V j). exact jH.
apply Sym. apply modify_seq_modifies_one_elt. unfold not.
intro H'. apply H. apply Sym. exact H'. Qed.

(* Specifications: *)
Lemma P6:forall(n:Nat)(G:Setoid)(A:part_set G)
(AH:seq(S n)A)(AH':full A =' seq_set AH)(AH'':distinct AH)
(c:Nat)(V:seq c(part_set G))
(Va:forall(i:fin c)(g:G),in_part g(V i)->in_part g A)
(Vb:forall(g:G)(H:in_part g A),exists i:fin c,in_part g(V i))
(Vc:forall(g:G)(H:in_part g A)(i i':fin c),
in_part g(V i)->in_part g(V i')->i =' i')
(V':seq c Nat)(V'a:forall(i:fin c),has_n_elements(V' i)(V i))
(i_:fin c)(i_H:in_part (subtype_elt(head AH))(V i_)),
forall(g:G)(H:in_part g (seq_set(Seqtl(Map_embed AH)))),
forall(i i':fin c),
in_part g((modify_seq V i_
(minus_part(V i_)(subtype_elt(head AH)))) i)->
in_part g((modify_seq V i_
(minus_part(V i_)(subtype_elt(head AH)))) i')->
i =' i'.