Section covering_1_B.
Hypothesis iB : IN X B.
Let stepB1 : EQ (EV F X) (EV FB X).
Proof restricted_to_th9 B F f X iB step7.
Let stepB2 : EQ (EV G X) (EV GB X).
Proof restricted_to_th9 B G g X iB step8.
Let stepB3 : EQ (EV GB X) (EV G X).
Proof EQ_symm (EV G X) (EV GB X) stepB2.
Let stepB4 : EQ (EV F X) (EV GB X).
Proof EQ_trans (EV F X) (EV FB X) (EV GB X) stepB1 step10.
Let stepB5 : EQ (EV F X) (EV G X).
Proof EQ_trans (EV F X) (EV GB X) (EV G X) stepB4 stepB3.
Lemma  (*Fact*) stepB6 : False.
Proof contra stepB5.
End covering_1_B.
Let step12 : ~ IN X B.
Proof stepB6.
Let step13 : ~ IN X U.
Proof union_th3 A B X step11 step12.
Lemma  (*Fact*) covering_em_Out : False.
Proof step13 iX.
End covering_em.
Proof excluded_middle (EQ (EV F X) (EV G X)) covering_em_Out.
End covering_1.

Theorem covering_th1 :
forall (A B F G : Ens) (f : is_a_function F) (g : is_a_function G)
(u : EQ (Domain F) (union A B))
(i : EQ (restricted_to A F) (restricted_to A G))
(j : EQ (restricted_to B F) (restricted_to B G)),
Extensionally_equivalent F G.