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 Th85:
  |[-1,0]|,|[1,0]| realize-max-dist-in D & p in LSeg(|[0,3]|,UMP D) implies
  (UMP D)`2 <= p`2
proof
  set x = UMP D;
  assume that
A1: a,b realize-max-dist-in D and
A2: p in LSeg(c,x);
A3: x in LSeg(c,x) by RLTOPSP1:68;
A4: LSeg(c,x) is vertical by A1,Th81;
A5: c = |[c`1,c`2]| by EUCLID:53;
A6: x = |[x`1,x`2]| by EUCLID:53;
  c in LSeg(c,x) by RLTOPSP1:68;
  then
A7: c`1 = x`1 by A3,A4;
  x`2 <= c`2 by A1,Lm21,Th83,JORDAN21:30;
  hence thesis by A2,A5,A6,A7,JGRAPH_6:1;
end;
