intros. elim(classic(i_=' j)).
intro H.
assert(gH':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 gH'. elim gH'. intros gH1 gH2. exact gH2.
intro H.
assert(gH':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 H1.
apply H. apply Sym. exact H1.
unfold not. intro gHEAD. apply H.
apply Vc with (g:=g)(i:=i_)(i':=j). apply (Va j). exact gH'.
apply in_part_comp_l with (subtype_elt(head AH)). exact i_H.
exact gHEAD. exact gH'. Qed.

(* Specifications: *)
Lemma P4helprr:forall(n:Nat)(G:Setoid)(A:part_set G)
(AH:seq(S n)A)(AH':full A =' seq_set AH)(AH'':distinct AH),
A =' seq_set(Map_embed AH).