reserve C for Simple_closed_curve,
  i for Nat;

theorem
  (Upper_Appr C).i is compact connected
proof
  Upper_Arc L~Cage(C,i) is compact;
  hence (Upper_Appr C).i is compact by JORDAN19:def 1;
  Upper_Arc L~Cage(C,i) is connected;
  hence thesis by JORDAN19:def 1;
end;
