Proof gr_Out.
Section glueing_domain1.
Variable F G X : Ens.
Hypothesis x : IN X (Domain (union F G)).
Let Y := EV (union F G) X.
Let step1 : IN (PAIR X Y) (union F G).
Proof EV_th1 (union F G) X x.
Section gd1a.
Hypothesis p1 : IN (PAIR X Y) F.
Let step2 : IN X (Domain F).
Proof Domain_th1 F X Y p1.

Lemma gd1a_Out : IN X (union (Domain F) (Domain G)).
Proof union_th1 (Domain F) (Domain G) X step2.