Section powerplus1.
Variable A X Y : Ens.
Hypothesis x : IN X A.
Hypothesis y : IN Y A.
Let P := PowerPlus A.
Let PP := PowerPlus P.
Let step1 : IN (First X) P.
Proof PowerPlus_th2 A (Singleton X) (Powerset_th4 A X x).
Let step2 : SUB P PP.
Proof PowerPlus_th1 P.
Let step3 : IN (First X) PP.
Proof step2 (First X) step1.
Let step4 : SUB (Powerset A) P.
Proof PowerPlus_th2 A.
Let step5 : SUB (Powerset (Powerset A)) (Powerset P).
Proof Powerset_th7 (Powerset A) P step4.
Let step6 : SUB (Powerset P) PP.
Proof PowerPlus_th2 P.
Let step7 : SUB (Powerset (Powerset A)) PP.
Proof SUB_trans (Powerset (Powerset A)) (Powerset P) PP step5 step6.
Let step8 : IN (Second Y) (Powerset (Powerset A)).
Proof Powerset_th6 A Y y.
Let step9 : IN (Second Y) PP.
Proof step7 (Second Y) step8.
Proof Powerset_th5 PP (First X) (Second Y) step3 step9.
End powerplus1.
Definition PowerTotal (A : Ens) := Powerset (PowerPlus (PowerPlus A)).

Theorem PowerTotal_th1 :
forall (A X Y : Ens) (x : IN X A) (y : IN Y A), IN (PAIR X Y) (PowerTotal A).