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;

theorem
  Fifth(MS,frequency) is_Between frequency,Octave(MS,frequency)
  proof
    consider fr be positive Real such that
A1: frequency = fr & Fifth(MS,frequency) = (3 qua Real) / 2 * fr by Def12;
A2: ex fr be positive Real st frequency = fr &
       Octave(MS,frequency) = 2 * fr by Def15;
    reconsider x = (3 qua Real) / 2 * fr as positive Real;
    1 * fr <= x < 2 * fr by XREAL_1:68;
    hence thesis by A2,A1;
  end;
