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;
reserve       HPS for Heptatonic_Pythagorean_Score,
        frequency for Element of HPS;

theorem
  for HPS being satisfying_euclidean Heptatonic_Pythagorean_Score
  for frequency being Element of HPS holds
  heptatonic_pythagorean_scale(HPS,frequency) is perfect_fifth
  proof
    let HPS be satisfying_euclidean Heptatonic_Pythagorean_Score;
    let frequency be Element of HPS;
A1:  heptatonic_pythagorean_scale(HPS,frequency) is heptatonic by Th88BIS;
    set gamme = heptatonic_pythagorean_scale(HPS,frequency);
A2: hepta_fondamental(HPS,frequency) = 1 * @frequency &
    hepta_1(HPS,frequency) = (9 qua Real) / 8 * @frequency &
    hepta_2(HPS,frequency) = (81 qua Real) / 64 * @frequency &
    hepta_3(HPS,frequency) = (4 qua Real) / 3 * @frequency &
    hepta_4(HPS,frequency) = (3 qua Real) / 2 * @frequency &
    hepta_5(HPS,frequency) = (27 qua Real) / 16 * @frequency &
    hepta_6(HPS,frequency) = (243 qua Real) / 128 * @frequency by Th88;
A3: ex fo be positive Real st frequency = fo &
    Octave(HPS,frequency) = 2 * fo by Def15;
    then hepta_octave(HPS,frequency) = 2 * @frequency;
    then
A4: hepta_octave(HPS,frequency) = gamme.8 by Th88;
A5: (the Ratio of HPS).(hepta_fondamental(HPS,frequency),
    hepta_4(HPS,frequency))
      = @hepta_4(HPS,frequency) / @hepta_fondamental(HPS,frequency) by Def25
     .= (3 qua Real) / 2 by A2,XCMPLX_1:89;
A6: (the Ratio of HPS).(hepta_1(HPS,frequency),
    hepta_5(HPS,frequency))
      = @hepta_5(HPS,frequency) / @hepta_1(HPS,frequency) by Def25
     .= (((27 qua Real) / 16)/((9 qua Real) / 8)) * (@frequency / @frequency)
       by A2,XCMPLX_1:76
     .= (3 qua Real) / 2 by XCMPLX_1:88;
A7: (the Ratio of HPS).(hepta_2(HPS,frequency),
       hepta_6(HPS,frequency))
      = @hepta_6(HPS,frequency) / @hepta_2(HPS,frequency) by Def25
     .= (((243 qua Real) / 128)/((81 qua Real) / 64)) *
        (@frequency / @frequency) by A2,XCMPLX_1:76
     .= (3 qua Real) / 2 by XCMPLX_1:88;
A8: (the Ratio of HPS).(hepta_3(HPS,frequency),
       hepta_octave(HPS,frequency))
      = @hepta_octave(HPS,frequency) / @hepta_3(HPS,frequency) by Def25
     .= ((2 qua Real) * @frequency) / ((4 qua Real) / 3 * @frequency)
       by Th88,A3
     .= ((2 qua Real) /((4 qua Real) / 3)) * (@frequency / @frequency)
       by XCMPLX_1:76
     .= (3 qua Real) / 2 by XCMPLX_1:88;
    reconsider n2 = 2,n3 = 3 as Element of HPS by Th20;
    (the Ratio of HPS).(n2,n3) = @n3 / @n2 by Def25
                              .= (3 qua Real) / 2;
    then n2,n3 equiv hepta_fondamental(HPS,frequency),hepta_4(HPS,frequency) &
      n2,n3 equiv hepta_1(HPS,frequency),hepta_5(HPS,frequency) &
      n2,n3 equiv hepta_2(HPS,frequency),hepta_6(HPS,frequency) &
      n2,n3 equiv hepta_3(HPS,frequency),hepta_octave(HPS,frequency)
      by A5,A6,A7,A8,Def08a;
    then [hepta_fondamental(HPS,frequency),hepta_4(HPS,frequency)]
      in fifth(HPS) &
      [hepta_1(HPS,frequency),hepta_5(HPS,frequency)] in fifth(HPS) &
      [hepta_2(HPS,frequency),hepta_6(HPS,frequency)] in fifth(HPS) &
      [hepta_3(HPS,frequency),hepta_octave(HPS,frequency)] in fifth(HPS)
      by EQREL_1:18;
    hence thesis by A4,A1,Def27;
  end;
