reserve C for Simple_closed_curve,
  i for Nat;
reserve R for non empty Subset of TOP-REAL 2,
  j, k, m, n for Nat;

theorem
  0 < n implies UMP L~Cage(C,n) = |[ (E-bound L~Cage(C,n) + W-bound L~
Cage(C,n)) / 2, upper_bound (proj2.:(L~Cage(C,n) /\
 LSeg(Gauge(C,n)*(Center Gauge(C,n),
  1), Gauge(C,n)*(Center Gauge(C,n),len Gauge(C,n))))) ]| by Th13;
