Section cartpair.
Variable A B X Y : Ens.
Hypothesis i : IN (PAIR X Y) (Cartesian A B).
Let U := PAIR X Y.
Let X1 := PR1 U.
Let Y1 := PR2 U.
Let step1 : EQ X X1.
Proof PAIR_proj_th3 X Y.
Let step2 : IN X1 A.
Proof Cartesian_proj1_th A B U i.
Proof Extensionality X X1 A step1 step2.
Let step3 : EQ Y Y1.
Proof PAIR_proj_th4 X Y.
Let step4 : IN Y1 B.
Proof Cartesian_proj2_th A B U i.
Proof Extensionality Y Y1 B step3 step4.
End cartpair.
Proof cartpairA_Out.

Theorem Cartesian_pair_th2 :
forall (A B X Y : Ens) (i : IN (PAIR X Y) (Cartesian A B)), IN Y B.