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 Th18:
  for C being compact Subset of TOP-REAL 2 holds p in BDD C
  implies South-Bound(p,C)`2 < p`2 & p`2 < North-Bound(p,C)`2
proof
  let C be compact Subset of TOP-REAL 2;
  assume
A1: p in BDD C;
  then South-Bound(p,C) in C & South-Bound(p,C) in south_halfline p by Th17;
  then South-Bound(p,C) in C /\ south_halfline p by XBOOLE_0:def 4;
  then
A2: proj2.:(C /\ south_halfline p) is non empty by Lm2,RELAT_1:119;
A3: BDD C misses C by JORDAN1A:7;
A4: now
A5: South-Bound(p,C)`1 = p`1 by EUCLID:52;
    assume South-Bound(p,C)`2 = p`2;
    then South-Bound(p,C) = p by A5,TOPREAL3:6;
    then p in C by A1,Th17;
    hence contradiction by A1,A3,XBOOLE_0:3;
  end;
  North-Bound(p,C) in C & North-Bound(p,C) in north_halfline p by A1,Th17;
  then C /\ north_halfline p is non empty by XBOOLE_0:def 4;
  then
A6: proj2.:(C /\ north_halfline p) is non empty by Lm2,RELAT_1:119;
  proj2.:(south_halfline p) is bounded_above & C /\ south_halfline p c=
  south_halfline p by Th4,XBOOLE_1:17;
  then
A7: upper_bound(proj2.:(C /\ south_halfline p)) <=
 upper_bound(proj2.:(south_halfline p) )
  by A2,RELAT_1:123,SEQ_4:48;
A8: now
A9: North-Bound(p,C)`1 = p`1 by EUCLID:52;
    assume North-Bound(p,C)`2 = p`2;
    then North-Bound(p,C) = p by A9,TOPREAL3:6;
    then p in C by A1,Th17;
    hence contradiction by A1,A3,XBOOLE_0:3;
  end;
  South-Bound(p,C)`2 = upper_bound(proj2.:(C /\ south_halfline p))
   & upper_bound(proj2.:(
  south_halfline p)) = p`2 by Th8,EUCLID:52;
  hence South-Bound(p,C)`2 < p`2 by A7,A4,XXREAL_0:1;
  proj2.:(north_halfline p) is bounded_below & C /\ north_halfline p c=
  north_halfline p by Th3,XBOOLE_1:17;
  then
A10: lower_bound(proj2.:(C /\ north_halfline p)) >=
 lower_bound(proj2.:(north_halfline p ))
  by A6,RELAT_1:123,SEQ_4:47;
  lower_bound(proj2.:(north_halfline p)) = p`2 &
   North-Bound(p,C)`2 = lower_bound(proj2.:
  (C /\ north_halfline p)) by Th7,EUCLID:52;
  hence thesis by A10,A8,XXREAL_0:1;
end;
