Section power1.
Variable A X Y : Ens.
Hypothesis x : IN X (Powerset A).
Hypothesis y : IN Y (Powerset A).
Let step1 : SUB X A.
Proof Powerset_pr2 A X x.
Let step2 : SUB Y A.
Proof Powerset_pr2 A Y y.
Let step3 : SUB (union X Y) A.
Proof union_pr3 X Y A step1 step2.
Proof Powerset_pr1 A (union X Y) step3.
Let step4 : SUB (intersection X Y) X.
Proof intersection_th1 X Y.
Let step5 : SUB (intersection X Y) A.
Proof SUB_trans (intersection X Y) X A step4 step1.
Proof Powerset_pr1 A (intersection X Y) step5.
End power1.
Proof power1_Out1.

Theorem Powerset_th3 :
forall (A X Y : Ens) (x : IN X (Powerset A)),
IN (intersection X Y) (Powerset A).