Section pair_proj2_proof.
Variable X Y : Ens.
Let M := PAIR X Y.
Let step1 : is_a_pair M.
Proof a_pair_is_a_pair X Y.
Let step2 : EQ M (PAIR (PR1 M) (PR2 M)).
Proof PAIR_proj_th1 M step1.
Proof PAIR_uniqueness X Y (PR1 M) (PR2 M) step2.
End pair_proj2_proof.

Theorem PAIR_proj_th2 :
forall X Y : Ens, pairwise_EQ X Y (PR1 (PAIR X Y)) (PR2 (PAIR X Y)).