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 Th88:
  |[-1,0]|,|[1,0]| realize-max-dist-in D implies
  LSeg(LMP D,|[0,-3]|) c= south_halfline LMP D
proof
  set p = LMP D;
  assume
A1: a,b realize-max-dist-in D;
  let x be object;
  assume
A2: x in LSeg(p,d);
  then reconsider x as Point of T2;
A3: p in LSeg(p,d) by RLTOPSP1:68;
A4: LSeg(p,d) is vertical by A1,Th82;
  then
A5: x`1 = p`1 by A2,A3;
A6: d = |[d`1,d`2]| by EUCLID:53;
A7: p = |[p`1,p`2]| by EUCLID:53;
  d in LSeg(p,d) by RLTOPSP1:68;
  then
A8: d`1 = p`1 by A3,A4;
  d`2 <= p`2 by A1,Lm23,Th84,JORDAN21:31;
  then x`2 <= p`2 by A2,A6,A7,A8,JGRAPH_6:1;
  hence thesis by A5,TOPREAL1:def 12;
end;
