theorem Th12:
  for C being compact Subset of TOP-REAL 2 holds
  north_halfline UMP C \ {UMP C} c= UBD C
proof
  let C be compact Subset of TOP-REAL 2;
  set A = north_halfline UMP C \ {UMP C};
  reconsider A as non bounded Subset of T2 by JORDAN2C:122,TOPREAL6:90;
  A is convex by JORDAN21:6;
  hence thesis by Th10,JORDAN2C:125;
end;
