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 Th82:
  |[-1,0]|,|[1,0]| realize-max-dist-in P implies
  LSeg(LMP P,|[0,-3]|) is vertical
proof
  assume a,b realize-max-dist-in P;
  then d`1 = (W-bound P + E-bound P) / 2 by Lm88
    .= (LMP P)`1 by EUCLID:52;
  hence thesis by SPPOL_1:16;
end;
