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 Th84:
  spiral_of_fifths(HPS,frequency,Fourth(HPS,frequency)).3
    = (9 qua Real) / 8 * @(frequency)
  proof
    set MS = HPS;
    set q = Fourth(MS,frequency);
    reconsider n1 = 2 as Nat;
    spiral_of_fifths(MS,frequency,q).n1 is Element of MS;
    then reconsider r32 = (3 qua Real) / 2 * @frequency as Element of MS
      by Th83;
A1: spiral_of_fifths(MS,frequency,q).3
      = spiral_of_fifths(MS,frequency,q).(n1 + 1)
     .= Fifth_reduct(MS,frequency,
           spiral_of_fifths(MS,frequency,q).n1) by Def19
     .= Fifth_reduct(MS,frequency,r32) by Th83;
    consider r,s be positive Real such that
A2: r = r32 & s = (3 qua Real) / 2 * r &
    Fifth(MS,r32) = s by Th54;
A3: 2 * @frequency < ((9 qua Real) / 4) * @frequency
      by XREAL_1:68;
A4: ex r being positive Real st Fifth(MS,r32) = r &
    Octave_descendent(MS,Fifth(MS,r32)) = r / 2 by Th51;
    not Fifth(MS,r32) is_Between frequency,Octave(MS,frequency)
    proof
      assume
A5:   Fifth(MS,r32) is_Between frequency,Octave(MS,frequency);
      ex fr be positive Real st frequency = fr &
      Octave(MS,frequency) = 2 * fr by Def15;
      hence contradiction by A5,A2,A3;
    end;
    hence thesis by A1,A2,A4,Def18;
  end;
