Proof
fun F G : Ens =>
Build_EQ (Domain (union F G)) (union (Domain F) (Domain G))
(glueing_th2 F G) (glueing_th3 F G).
Section glueing_proof.
Variable F G : Ens.
Hypothesis f : is_a_function F.
Hypothesis g : is_a_function G.
Hypothesis e : Extensionally_equivalent F G.
Let H := union F G.
Let step1 : is_a_relation F.
Proof function_is_a_relation F f.
Let step2 : is_a_relation G.
Proof function_is_a_relation G g.
Let step3 : well_definedness F.
Proof function_well_definedness F f.
Let step4 : well_definedness G.
Proof function_well_definedness G g.
Let step5 : is_a_relation H.
Proof glueing_th1 F G step1 step2.
Section glu_wd.
Variable X Y Z : Ens.
Hypothesis i : IN (PAIR X Y) H.
Hypothesis j : IN (PAIR X Z) H.
Section gluA.
Hypothesis j1 : IN (PAIR X Z) F.
Hypothesis i1 : IN (PAIR X Y) F.

Lemma gluA_Out : EQ Y Z.
Proof step3 X Y Z i1 j1.