Section COMPp.
Variable F G X Y : Ens.
Hypothesis i : IN (PAIR X Y) (COMP F G).
Let Z := PAIR X Y.
Let step1 : IN Z (COMP F G).
Proof i.
Let step2 : EQ X (PR1 Z).
Proof PAIR_proj_th3 X Y.
Let step3 : EQ Y (PR2 Z).
Proof PAIR_proj_th4 X Y.
Let step4 : IN (PAIR (PR1 Z) (Comp_INTERp F G X Y)) F.
Proof COMP_th3 F G Z step1.
Let step5 : IN (PAIR (Comp_INTERp F G X Y) (PR2 Z)) G.
Proof COMP_th4 F G Z step1.
Proof
Substitute X (PR1 Z) (fun U : Ens => IN (PAIR U (Comp_INTERp F G X Y)) F)
step2 step4.
Proof
Substitute Y (PR2 Z) (fun U : Ens => IN (PAIR (Comp_INTERp F G X Y) U) G)
step3 step5.
End COMPp.

Theorem COMP_th3p :
forall (F G X Y : Ens) (i : IN (PAIR X Y) (COMP F G)),
IN (PAIR X (Comp_INTERp F G X Y)) F.