Section comp_ev_3.
Variable F G X : Ens.
Hypothesis f : is_a_function F.
Hypothesis g : is_a_function G.
Hypothesis i : IN X (Domain F).
Hypothesis j : IN (EV F X) (Domain G).
Let Z := EV G (EV F X).
Let W := EV (COMP F G) X.
Let step1 : is_a_function (COMP F G).
Proof composition_th2 F G f g.
Let step2 : well_definedness (COMP F G).
Proof function_well_definedness (COMP F G) step1.
Let step3 : IN (PAIR X Z) (COMP F G).
Proof composition_th3 F G X i j.
Let step4 : IN X (Domain (COMP F G)).
Proof Domain_th1 (COMP F G) X Z step3.
Let step5 : IN (PAIR X W) (COMP F G).
Proof EV_th1 (COMP F G) X step4.
Proof step2 X W Z step5 step3.
End comp_ev_3.

Theorem composition_th4 :
forall (F G X : Ens) (f : is_a_function F) (g : is_a_function G)
(i : IN X (Domain F)) (j : IN (EV F X) (Domain G)),
EQ (EV (COMP F G) X) (EV G (EV F X)).