End gluB.
Section gluC.
Hypothesis j1 : IN (PAIR X Z) F.
Hypothesis i1 : IN (PAIR X Y) G.
Let Cstep1 : IN X (Domain F).
Proof Domain_th1 F X Z j1.
Let Cstep2 : IN X (Domain G).
Proof Domain_th1 G X Y i1.
Let Cstep3 : EQ Y (EV G X).
Proof function_th1 G g X Y i1.
Let Cstep4 : EQ Z (EV F X).
Proof function_th1 F f X Z j1.
Let Cstep5 : EQ (EV F X) Z.
Proof EQ_symm Z (EV F X) Cstep4.
Let Cstep6 : EQ (EV F X) (EV G X).
Proof e X Cstep1 Cstep2.
Let Cstep6a : EQ (EV G X) (EV F X).
Proof EQ_symm (EV F X) (EV G X) Cstep6.
Let Cstep7 : EQ Y (EV F X).
Proof EQ_trans Y (EV G X) (EV F X) Cstep3 Cstep6a.

(* Specifications: *)
Lemma gluC_Out : EQ Y Z.
Proof EQ_trans Y (EV F X) Z Cstep7 Cstep5.