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 Th85:
  spiral_of_fifths(HPS,frequency,Fourth(HPS,frequency)).4
    = (27 qua Real) / 16 * @frequency
  proof
    set MS = HPS;
    reconsider n2 = 3 as Nat;
    set q = Fourth(MS,frequency);
    spiral_of_fifths(MS,frequency,q).n2 is Element of MS;
    then reconsider r32 = (9 qua Real) / 8 * @frequency as Element of MS
      by Th84;
A1: spiral_of_fifths(MS,frequency,q).4
      = spiral_of_fifths(MS,frequency,q).(n2 + 1)
     .= Fifth_reduct(MS,frequency,
            spiral_of_fifths(MS,frequency,q).n2) by Def19
     .= Fifth_reduct(MS,frequency,r32) by Th84;
    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;
    ((27 qua Real) / 16) * @frequency < 2 * @frequency &
      1 * @frequency < ((27 qua Real)/16) * @frequency by XREAL_1:68;
    then Fifth(MS,r32) is_Between frequency,Octave(MS,frequency)
      by A2,A3;
    hence thesis by A2,A1,Def18;
  end;
