intros. simpl. unfold Map_eq. intro j. destruct i. destruct j.
rename index into i. rename index0 into j.
rename in_range_prf into iH. rename in_range_prf0 into jH.
elim(classic(i=j)). intro H.
apply Trans with (modify_seq(modify_seq Q(Build_finiteT jH)z)
(Build_finiteT jH) (Q(Build_finiteT jH)) (Build_finiteT jH)).
apply modify_comp. apply Map_compatible_prf. auto.
apply modify_comp. apply Refl. apply Refl. auto. auto.
apply modify_seq_defprop. intro H.
apply Trans with
(modify_seq Q(Build_finiteT iH)z(Build_finiteT jH)).
apply modify_seq_modifies_one_elt. auto.
apply modify_seq_modifies_one_elt. auto. Qed.

(* Specifications: *)
Lemma P4helpr: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_))
(j:fin c)(g:G)(gH:in_part g (
(modify_seq V i_(minus_part(V i_)(subtype_elt(head AH))))j
)),~g =' head(Map_embed AH).