End gluA.
Section gluB.
Hypothesis j1 : IN (PAIR X Z) G.
Hypothesis i1 : IN (PAIR X Y) F.
Let Bstep1 : IN X (Domain F).
Proof Domain_th1 F X Y i1.
Let Bstep2 : IN X (Domain G).
Proof Domain_th1 G X Z j1.
Let Bstep3 : EQ Y (EV F X).
Proof function_th1 F f X Y i1.
Let Bstep4 : EQ Z (EV G X).
Proof function_th1 G g X Z j1.
Let Bstep5 : EQ (EV G X) Z.
Proof EQ_symm Z (EV G X) Bstep4.
Let Bstep6 : EQ (EV F X) (EV G X).
Proof e X Bstep1 Bstep2.
Let Bstep7 : EQ Y (EV G X).
Proof EQ_trans Y (EV F X) (EV G X) Bstep3 Bstep6.

Lemma gluB_Out : EQ Y Z.
Proof EQ_trans Y (EV G X) Z Bstep7 Bstep5.