intros. assert(HYP:Pred_fun(seq_set(Seqtl(Map_embed AH)))g). auto.
set(hx:=Build_subtype(P:=seq_set(Seqtl(Map_embed AH)))
(subtype_elt:=g)HYP). elim(has_index hx).
intros j jH. apply in_part_comp_l with ((Map_embed(Seqtl AH))j).
apply Map_embed_prop. apply Trans with (Seqtl(Map_embed AH)j).
exact jH. apply Sym. apply Map_embed_Seqtl. Qed.

(* Specifications: *)
Lemma seq_set_head_or_tail_inv:
forall(n:Nat)(G:Setoid)(A:part_set G)
(AH:seq(S n)A)(AH':full A =' seq_set AH)(AH'':distinct AH)
(g:G)(gH:in_part g (seq_set(Seqtl(Map_embed AH)))),
~g =' subtype_elt(head AH).