Proof
fun (A B F G : Ens) (f : is_a_function F) (g : is_a_function G)
(u : EQ (Domain F) (union A B)) (v : EQ (Domain G) (union A B))
(i : EQ (restricted_to A F) (restricted_to A G))
(j : EQ (restricted_to B F) (restricted_to B G)) =>
Extensionality_for_functions F G f g
(EQ_trans (Domain F) (union A B) (Domain G) u
(EQ_symm (Domain G) (union A B) v)) (covering_th1 A B F G f g u i j).
(***** glueing two functions ******)
Section glueing_relation_proof.
Variable F G : Ens.
Hypothesis f : is_a_relation F.
Hypothesis g : is_a_relation G.
Let H := union F G.
Variable X : Ens.
Hypothesis x : IN X H.
Section gr1.
Hypothesis x1 : IN X F.

(* Specifications: *)
Lemma gr1_Out : is_a_pair X.
Proof f X x1.