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 Th90:
  |[-1,0]|,|[1,0]| realize-max-dist-in C & P is_inside_component_of C implies
  LSeg(LMP C,|[0,-3]|) misses P
proof
  set m = LMP C;
  set L = LSeg(m,d);
  assume that
A1: a,b realize-max-dist-in C and
A2: P is_inside_component_of C;
A3: ex VP being Subset of T2|C` st ( VP = P)&( VP
is a_component)&( VP is bounded Subset of Euclid 2) by A2,JORDAN2C:13;
  m in L by RLTOPSP1:68;
  then {m} c= L by ZFMISC_1:31;
  then
A4: L = L \ {m} \/ {m} by XBOOLE_1:45;
A5: L \ {m} c= south_halfline m \ {m} by A1,Th88,XBOOLE_1:33;
  south_halfline m \ {m} c= UBD C by Th13;
  then L \ {m} c= UBD C by A5;
  then
A6: L \ {m} misses P by A2,Th14,XBOOLE_1:63;
  {m} misses P by A3,Lm4,JORDAN21:31;
  hence thesis by A4,A6,XBOOLE_1:70;
end;
