theorem Th81:
  |[-1,0]|,|[1,0]| realize-max-dist-in P implies
  LSeg(|[0,3]|,UMP P) is vertical
proof
  assume a,b realize-max-dist-in P;
  then c`1 = (W-bound P + E-bound P) / 2 by Lm87
    .= (UMP P)`1 by EUCLID:52;
  hence thesis by SPPOL_1:16;
end;
