reserve C for Simple_closed_curve,
  i for Nat;

theorem
  (Lower_Appr C).i is compact connected
proof
  Lower_Arc L~Cage(C,i) is compact;
  hence (Lower_Appr C).i is compact by JORDAN19:def 2;
  Lower_Arc L~Cage(C,i) is connected;
  hence thesis by JORDAN19:def 2;
end;
