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.

Lemma covering_1_Out : EQ (EV F X) (EV G X).