Proof one_different_from_two_Out.
(****************************************)
(**********     PAIR   ****************)
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Definition First (X : Ens) : Ens := Singleton X.
Definition Second (X : Ens) : Ens := Doubleton Zero (Singleton X).
Definition PAIR (X Y : Ens) : Ens := Doubleton (First X) (Second Y).
Section first_uniqueness.
Variable A : Ens.
Variable B : Ens.
Hypothesis m : EQ (First A) (First B).
Let F := First A.
Let G := First B.
Let FsubG : SUB F G := EQ_forwards F G m.
Let iAF : IN A F := Singleton_pr1 A.
Let iBF : IN A G := FsubG A iAF.
Let step1 : EQ B A := Singleton_pr2 B A iBF.
Definition first_uniqueness_Out : EQ A B := EQ_symm B A step1.
End first_uniqueness.

Theorem First_uniqueness :
forall (A B : Ens) (m : EQ (First A) (First B)), EQ A B.