reserve C for Simple_closed_curve,
  i for Nat;

theorem Th1:
  (Upper_Appr C).i c= Cl RightComp Cage(C,0)
proof
A1: Upper_Arc L~Cage (C,i) c= L~Cage (C,i) by JORDAN6:61;
A2: L~Cage (C,i) c= Cl RightComp Cage(C,0) by JORDAN1H:46;
  (Upper_Appr C).i = Upper_Arc L~Cage (C,i) by JORDAN19:def 1;
  hence thesis by A1,A2;
end;
