reserve n for Element of NAT,
  V for Subset of TOP-REAL n,
  s,s1,s2,t,t1,t2 for Point of TOP-REAL n,
  C for Simple_closed_curve,
  P for Subset of TOP-REAL 2,
  a,p ,p1,p2,q,q1,q2 for Point of TOP-REAL 2;

theorem Th19:
  for C being compact Subset of TOP-REAL 2 holds p in BDD C
implies lower_bound(proj2.:(C /\ north_halfline p)) >
 upper_bound(proj2.:(C /\ south_halfline p
  ))
proof
  let C be compact Subset of TOP-REAL 2;
  assume p in BDD C;
  then
A1: South-Bound(p,C)`2 < p`2 & p`2 < North-Bound(p,C)`2 by Th18;
  North-Bound(p,C)`2 = lower_bound(proj2.:(C /\ north_halfline p))
   & South-Bound(p
  ,C) `2 = upper_bound(proj2.:(C /\ south_halfline p)) by EUCLID:52;
  hence thesis by A1,XXREAL_0:2;
end;
