Section Subsets.
Variable a : A0.
(* We define the type of elements of A0 smaller than a w.r.t embedding. The Record is in Type, but it is possible to avoid such structure. *)
Record sub : Type :=  {witness : A0; emb_wit : emb witness a}.
(* F is its image through i0 *)
Definition F : A0 := i0 sub (Rof _ _ emb witness).

(* F is embedded in Omega: - the witness projection is a morphism - a is an upper bound because emb_wit proves that witness is smaller than a. *)
Lemma F_emb_Omega : emb F Omega.