(** The paradoxical set *)
(** Bijection between [U] and [(pow U)] *)
Definition f (u:U) : pow U := u U.
Definition g (h:pow U) : U :=
fun X => let lX := j2 (L1 X U) in let rU := i2 (L1 U U) in lX (rU h).

(** We deduce that the powerset of [U] is a retract of [U]. This lemma is stated in Berardi's article, but is not used afterwards. *)
Lemma retract_pow_U_U : retract (pow U) U.
Definition Not_b (b:Bool) := IFProp (b = T) F T.