Section comp_ev_2.
Variable F G X : Ens.
Hypothesis i : IN X (Domain F).
Hypothesis j : IN (EV F X) (Domain G).
Let Y := EV F X.
Let Z := EV G Y.
Let step1 : IN (PAIR X Y) F.
Proof EV_th1 F X i.
Let step2 : IN (PAIR Y Z) G.
Proof EV_th1 G Y j.
Proof COMP_th5 F G X Y Z step1 step2.
End comp_ev_2.

Theorem composition_th3 :
forall (F G X : Ens) (i : IN X (Domain F)) (j : IN (EV F X) (Domain G)),
IN (PAIR X (EV G (EV F X))) (COMP F G).