reserve m,n,i,i2,j for Nat,
  r,r1,r2,s,t for Real,
  x,y,z for object;
reserve p,p1,p2,p3,q,q1,q2,q3,q4 for Point of TOP-REAL n;
reserve u for Point of Euclid n;
reserve R for Subset of TOP-REAL n;
reserve P,Q for Subset of TOP-REAL n;
reserve D for non vertical non horizontal non empty compact Subset of TOP-REAL
  2;
reserve f for clockwise_oriented non constant standard
  special_circular_sequence;
reserve p for Point of TOP-REAL 2;

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;
