Section gluD.
Hypothesis j1 : IN (PAIR X Z) G.
Hypothesis i1 : IN (PAIR X Y) G.
Proof step4 X Y Z i1 j1.
End gluD.
Let step6 : IN (PAIR X Y) F -> EQ Y Z.
Proof
union_th4 (IN (PAIR X Y) F -> EQ Y Z) F G (PAIR X Z) j gluA_Out gluB_Out.
Let step7 : IN (PAIR X Y) G -> EQ Y Z.
Proof
union_th4 (IN (PAIR X Y) G -> EQ Y Z) F G (PAIR X Z) j gluC_Out gluD_Out.
Proof union_th4 (EQ Y Z) F G (PAIR X Y) i step6 step7.
End glu_wd.
Let step8 : well_definedness H.
Proof glu_wd_Out.
Proof Build_is_a_function H step5 step8.
End glueing_proof.

Theorem glueing_th5 :
forall (F G : Ens) (f : is_a_function F) (g : is_a_function G)
(e : Extensionally_equivalent F G), is_a_function (union F G).