theorem Th81:
  Fourth(HPS,frequency) is_Between frequency,Octave(HPS,frequency)
  proof
    set MS = HPS;
    consider fr be positive Real such that
A1: frequency = fr and
A2: Fourth(MS,frequency) = (4 qua Real) / 3 * fr by Def24;
    consider fr2 be positive Real such that
A3: frequency = fr2 and
A4: Octave(MS,frequency) = 2 * fr2 by Def15;
    1 * fr <= (4 qua Real) / 3 * fr < 2 * fr by XREAL_1:68;
    hence thesis by A3,A4,A1,A2;
  end;
