Section efdom.
Variable X : Ens.
Hypothesis i : IN X (Domain Zero).
Let Y := EV Zero X.
Let step1 : IN (PAIR X Y) Zero.
Proof EV_th1 Zero X i.
Proof EmptySet_th1 (PAIR X Y) step1.
End efdom.
Proof efdom_Out.
Proof
EQ_symm (Domain Zero) Zero
(its_empty_implies_equals_emptyset (Domain Zero)
empty_function_domain_empty).
(********** Cartesian product *********)
Definition PowerPlus (A : Ens) := union A (Powerset A).
Proof fun A : Ens => union_pr1 A (Powerset A).

Theorem PowerPlus_th2 : forall A : Ens, SUB (Powerset A) (PowerPlus A).