intros. rename i into j. elim(classic(i_ =' j)). intro H.
apply has_n_elements_comp with (n:=V' i_-1)
(B:=modify_seq V i_(minus_part(V i_)(subtype_elt(head AH)))i_).
apply has_n_elements_comp with (n:=V' i_-1)
(B:=minus_part(V i_)(subtype_elt(head AH))).
apply shrink_one. exact (V'a i_). exact i_H.
apply Sym. apply modify_seq_defprop. apply Refl.
apply Map_compatible_prf. exact H.
apply Trans with (modify_seq V' i_(V' i_-1)i_).
apply Sym. apply modify_seq_defprop. apply Map_compatible_prf.
exact H.
intro H. apply has_n_elements_comp with (n:=V' j)
(B:=modify_seq V i_(minus_part(V i_)(subtype_elt(head AH)))j).
apply has_n_elements_comp with (n:=V' j) (B:=V j).
exact (V'a j). apply Sym. apply modify_seq_modifies_one_elt.
unfold not. intro H'. apply H. apply Sym. exact H'.
apply Refl. apply Refl. apply Sym.
apply modify_seq_modifies_one_elt.
unfold not. intro H'. apply H. apply Sym. exact H'. Qed.

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