reserve a, b, c, d, r, s for Real,
  n for Element of NAT,
  p, p1, p2 for Point of TOP-REAL 2,
  x, y for Point of TOP-REAL n,
  C for Simple_closed_curve,
  A, B, P for Subset of TOP-REAL 2,
  U, V for Subset of (TOP-REAL 2)|C`,
  D for compact with_the_max_arc Subset of TOP-REAL 2;

theorem Th11:
  for C being compact Subset of TOP-REAL 2 holds
  south_halfline LMP C \ {LMP C} misses C
proof
  let C be compact Subset of TOP-REAL 2;
  set p = LMP C;
  set L = south_halfline p;
  set w = (W-bound C + E-bound C) / 2;
  assume L \ {p} meets C;
  then consider x being object such that
A1: x in L \ {p} and
A2: x in C by XBOOLE_0:3;
A3: x in L by A1,ZFMISC_1:56;
A4: x <> p by A1,ZFMISC_1:56;
  reconsider x as Point of T2 by A1;
A5: x`1 = p`1 by A3,TOPREAL1:def 12;
A6: x`2 <= p`2 by A3,TOPREAL1:def 12;
  x`2 <> p`2 by A4,A5,TOPREAL3:6;
  then
A7: x`2 < p`2 by A6,XXREAL_0:1;
  x`1 = w by A5,EUCLID:52;
  then x in Vertical_Line w by JORDAN6:31;
  then x in C /\ Vertical_Line w by A2,XBOOLE_0:def 4;
  hence thesis by A7,JORDAN21:29;
end;
