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 & q in BDD C
  & p`1 <> q`1 implies North-Bound(p,C), South-Bound(q,C), North-Bound(q,C),
  South-Bound(p,C) are_mutually_distinct
proof
  let C be compact Subset of TOP-REAL 2;
  set np = North-Bound(p,C), sq = South-Bound(q,C), nq = North-Bound(q,C), sp
  = South-Bound(p,C);
A1: np`1 = p`1 & sp`1 = p`1 by EUCLID:52;
A2: North-Bound(q,C)`2 = lower_bound(proj2.:(C /\ north_halfline q)) &
South-Bound(q
  ,C) `2 = upper_bound(proj2.:(C /\ south_halfline q)) by EUCLID:52;
A3: 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 p in BDD C & q in BDD C & p`1 <> q`1;
  hence np <> sq & np <> nq & np <> sp & sq <> nq & sq <> sp & nq <> sp by A1
,A2,A3,Th19,EUCLID:52;
end;
