Section total_and_pairs_1.
Variable R : Ens.
Variable A B : Ens.
Hypothesis i : IN (PAIR A B) R.
Let step1 : IN A (First A).
Proof Singleton_pr1 A.
Let step2 : IN B (Singleton B).
Proof Singleton_pr1 B.
Let step3 : IN (Singleton B) (Second B).
Proof Doubleton_th2 Zero (Singleton B).
Let step4 : IN (First A) (PAIR A B).
Proof Doubleton_th1 (First A) (Second B).
Let step5 : IN (Second B) (PAIR A B).
Proof Doubleton_th2 (First A) (Second B).
Proof Total_th3 A (First A) (PAIR A B) R step1 step4 i.
Proof Total_th4 B (Singleton B) (Second B) (PAIR A B) R step2 step3 step5 i.
End total_and_pairs_1.
Proof total_and_pairs_1_OutA.

Theorem Relation_Total_th2 :
forall (R A B : Ens) (i : IN (PAIR A B) R), IN B (Total R).