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 Th80:
  for MS being satisfying_fourth_constructible MusicSpace
  st MS = REAL_Music holds
  for frequency being Element of MS
  ex fr being positive Real st
  frequency = fr & Fourth(MS,frequency) = (4 qua Real) / 3 * fr
  proof
    let MS be satisfying_fourth_constructible MusicSpace;
    assume
A1: MS = REAL_Music;
    let frequency be Element of MS;
    reconsider fr = frequency as positive Real by A1,Th1;
    take fr;
    thus frequency = fr;
    consider fr9,qr9 be positive Real such that
A2: fr9 = frequency and
A3: qr9 = (4 qua Real) / 3 * fr9 and
A4: [fr9,qr9] in fourth(MS) by A1,Th78;
    reconsider qr9 as Element of MS by A1,Th1;
      qr9 = Fourth(MS,frequency) by A2,A4,Def23;
    hence thesis by A2,A3;
  end;
