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 Th23:
  0 < n implies (UMP C)`2 < (UMP Upper_Arc L~Cage(C,n))`2
proof
  assume 0 < n;
  then UMP Upper_Arc L~Cage(C,n) = UMP L~Cage(C,n) by Th21;
  hence thesis by Th17;
end;
