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
  for C being compact Subset of TOP-REAL 2 holds p in BDD C implies
  South-Bound(p,C) <> North-Bound(p,C)
proof
  let C be compact Subset of TOP-REAL 2;
  assume
A1: p in BDD C;
A2: 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;
  assume not thesis;
  hence thesis by A1,A2,Th19;
end;
