reserve MS for satisfying_equiv MusicStruct;
reserve a,b,c,d,e,f for Element of MS;
reserve MS for satisfying_octave_descendent_constructible
  classical_octave satisfying_octave_constructible classical_fifth
  satisfying_fifth_constructible satisfying_harmonic_closed satisfying_Nat
  satisfying_commutativity satisfying_interval satisfying_equiv satisfying_Real
  non empty MusicStruct,
  fondamentale,frequency for Element of MS;
reserve                              MS for MusicSpace,
        fondamentale, frequency, f1, f2 for Element of MS;
reserve       HPS for Heptatonic_Pythagorean_Score,
        frequency for Element of HPS;

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;
