Section comp_ev_1.
Variable F G X Y Z : Ens.
Hypothesis i : IN (PAIR X Y) F.
Hypothesis j : IN (PAIR Y Z) G.
Let step1 : Comp_inter F G X Y Z.
Proof Build_Comp_inter F G X Y Z i j.
Let W := PAIR X Z.
Let step2 : EQ X (PR1 W).
Proof PAIR_proj_th3 X Z.
Let step3 : EQ (PR1 W) X.
Proof EQ_symm X (PR1 W) step2.
Let step4 : EQ Z (PR2 W).
Proof PAIR_proj_th4 X Z.
Let step5 : EQ (PR2 W) Z.
Proof EQ_symm Z (PR2 W) step4.
Let step6 : Comp_inter F G X Y (PR2 W).
Proof Substitute (PR2 W) Z (fun U : Ens => Comp_inter F G X Y U) step5 step1.
Let step7 : Comp_inter F G (PR1 W) Y (PR2 W).
Proof
Substitute (PR1 W) X (fun U : Ens => Comp_inter F G U Y (PR2 W)) step3
step6.
Let step8 : EXISTS (fun Y1 : Ens => Comp_inter F G (PR1 W) Y1 (PR2 W)).
Proof EXISTS_th1 (fun Y1 : Ens => Comp_inter F G (PR1 W) Y1 (PR2 W)) Y step7.
Let step9 : Comp_prop F G W.
Proof Build_Comp_prop F G W (a_pair_is_a_pair X Z) step8.
Proof COMP_pr1 F G W step9.
End comp_ev_1.

Theorem COMP_th5 :
forall (F G X Y Z : Ens) (i : IN (PAIR X Y) F) (j : IN (PAIR Y Z) G),
IN (PAIR X Z) (COMP F G).