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 Th37:
  UMP C <> LMP C
proof
  assume
A1: UMP C = LMP C;
  (LMP C)`2 < (UMP C)`2 by Th36;
  hence contradiction by A1;
end;
