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 Th86:
  spiral_of_fifths(HPS,frequency,Fourth(HPS,frequency)).5
    = (81 qua Real) / 64 * @frequency
  proof
    set MS = HPS;
    set q = Fourth(MS,frequency);
    reconsider n3 = 4 as Nat;
    spiral_of_fifths(MS,frequency,q).n3 is Element of MS;
    then reconsider r32 = (27 qua Real) / 16 * @frequency as Element of MS
    by Th85;
A1: spiral_of_fifths(MS,frequency,q).5
      = spiral_of_fifths(MS,frequency,q).(n3 + 1)
     .= Fifth_reduct(MS,frequency,
           spiral_of_fifths(MS,frequency,q).n3) by Def19
     .= Fifth_reduct(MS,frequency,r32) by Th85;
    consider r,s be positive Real such that
A2: r = r32 & s = (3 qua Real) / 2 * r &
      Fifth(MS,r32) = s by Th54;
A3: ex fr be positive Real st frequency = fr &
      Octave(MS,frequency) = 2 * fr by Def15;
A4: not @frequency <= ((81 qua Real)/32) * @frequency
                   <= 2 * @frequency by XREAL_1:68;
A5: 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) by A4,A2,A3;
    hence thesis by A1,A2,A5,Def18;
  end;
