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 Th86:
  |[-1,0]|,|[1,0]| realize-max-dist-in D & p in LSeg(LMP D,|[0,-3]|) implies
  p`2 <= (LMP D)`2
proof
  set x = LMP D;
  assume that
A1: a,b realize-max-dist-in D and
A2: p in LSeg(x,d);
A3: x in LSeg(x,d) by RLTOPSP1:68;
A4: LSeg(x,d) is vertical by A1,Th82;
A5: d = |[d`1,d`2]| by EUCLID:53;
A6: x = |[x`1,x`2]| by EUCLID:53;
  d in LSeg(x,d) by RLTOPSP1:68;
  then
A7: d`1 = x`1 by A3,A4;
  d`2 <= x`2 by A1,Lm23,Th84,JORDAN21:31;
  hence thesis by A2,A5,A6,A7,JGRAPH_6:1;
end;
