theorem
  for C being compact Subset of TOP-REAL 2 for p being Point of TOP-REAL
  2 st south_halfline p misses C holds south_halfline p c= UBD C
proof
  let C be compact Subset of TOP-REAL 2;
  let p be Point of TOP-REAL 2;
  set WH = south_halfline p;
  assume
A1: WH misses C;
  WH is non bounded non empty connected by Th107;
  hence thesis by A1,Th109;
end;
