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 Th82:
  spiral_of_fifths(HPS,frequency,Fourth(HPS,frequency)).1
    = frequency
  proof
    set MS = HPS;
    set  q = Fourth(MS,frequency),
        f1 = spiral_of_fifths(MS,frequency,q).1;
    consider frq be positive Real such that
A1: frequency = frq & Fourth(MS,frequency) = (4 qua Real) / 3 * frq
      by Def24;
    reconsider n1 = 1, n0 = 0 as Nat;
A2: spiral_of_fifths(MS,frequency,q).1
      = spiral_of_fifths(MS,frequency,q).(n0 + 1)
     .= Fifth_reduct(MS,frequency,
           spiral_of_fifths(MS,frequency,q).n0) by Def19
     .= Fifth_reduct(MS,frequency,q) by Def19;
    consider r,s be positive Real such that
A3: r = q & s = (3 qua Real) / 2 * r &
    Fifth(MS,q) = s by Th54;
    consider fr2 be positive Real such that
A4: frequency = fr2 and
A5: Octave(MS,frequency) = 2 * fr2 by Def15;
    consider ro be positive Real such that
A6: Fifth(MS,q) = ro and
A7: Octave_descendent(MS,Fifth(MS,q)) = ro / 2 by Th51;
    not Fifth(MS,q) is_Between frequency,Octave(MS,frequency)
      by A3,A1,A4,A5;
    hence thesis by A2,A6,A7,A3,A1,Def18;
  end;
