theorem Th13:
  for C being compact Subset of TOP-REAL 2 holds
  south_halfline LMP C \ {LMP C} c= UBD C
proof
  let C be compact Subset of TOP-REAL 2;
  set A = south_halfline LMP C \ {LMP C};
  reconsider A as non bounded Subset of T2 by JORDAN2C:123,TOPREAL6:90;
  A is convex by JORDAN21:7;
  hence thesis by Th11,JORDAN2C:125;
end;
