(** The powerset operator *)
Definition pow (P:Prop) := P -> Bool.Record retract : Prop :=
  {i : A -> B; j : B -> A; inv : forall a:A, j (i a) = a}.Record retract : Prop :=
  {i : A -> B; j : B -> A; inv : forall a:A, j (i a) = a}.Record retract : Prop :=
  {i : A -> B; j : B -> A; inv : forall a:A, j (i a) = a}.Record retract : Prop :=
  {i : A -> B; j : B -> A; inv : forall a:A, j (i a) = a}.Record retract_cond : Prop :=
  {i2 : A -> B; j2 : B -> A; inv2 : retract -> forall a:A, j2 (i2 a) = a}.Record retract_cond : Prop :=
  {i2 : A -> B; j2 : B -> A; inv2 : retract -> forall a:A, j2 (i2 a) = a}.Record retract_cond : Prop :=
  {i2 : A -> B; j2 : B -> A; inv2 : retract -> forall a:A, j2 (i2 a) = a}.Record retract_cond : Prop :=
  {i2 : A -> B; j2 : B -> A; inv2 : retract -> forall a:A, j2 (i2 a) = a}.

Lemma AC : forall r:retract_cond, retract -> forall a:A, j2 r (i2 r a) = a.