reserve C for Simple_closed_curve,
  P for Subset of TOP-REAL 2,
  p for Point of TOP-REAL 2,
  n for Element of NAT;
reserve D for compact with_the_max_arc Subset of TOP-REAL 2;

theorem
  Lower_Middle_Point C <> Upper_Middle_Point C
proof
  (Lower_Middle_Point C)`2 <> (Upper_Middle_Point C)`2 by Th15;
  hence thesis;
end;
