Proof powerplus2_Out.
Record in_cartesian (A B P : Ens) : Prop :=
{in_cartesian_pair : is_a_pair P;
in_cartesian_pr1 : IN (PR1 P) A;
in_cartesian_pr2 : IN (PR2 P) B}.
Section in_cartesian_bounded_proof.
Variable A B : Ens.
Section cpb1.
Variable U : Ens.
Hypothesis i : in_cartesian A B U.
Let step1 : is_a_pair U.
Proof in_cartesian_pair A B U i.
Let X := PR1 U.
Let Y := PR2 U.
Let step4 : EQ U (PAIR X Y).
Proof PAIR_proj_th1 U step1.
Let step5 : IN X A.
Proof in_cartesian_pr1 A B U i.
Let step6 : IN Y B.
Proof in_cartesian_pr2 A B U i.
Let step7 : IN (PAIR X Y) (PowerTotal (union A B)).
Proof PowerTotal_th2 A B X Y step5 step6.

Lemma cpb1_Out : IN U (PowerTotal (union A B)).
Proof Extensionality U (PAIR X Y) (PowerTotal (union A B)) step4 step7.