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;

theorem Th59:
  spiral_of_fifths(MS,fondamentale,fondamentale).2
    = (9 qua Real) / 8 * @fondamentale
  proof
    reconsider n2 = 2,n1 = 1,n0 = 0 as Nat;
    spiral_of_fifths(MS,fondamentale,fondamentale).n1 is Element of MS;
    then reconsider r32 = (3 qua Real) / 2 * @fondamentale as Element of MS
      by Th58;
A1: spiral_of_fifths(MS,fondamentale,fondamentale).2
      = spiral_of_fifths(MS,fondamentale,fondamentale).(n1 + 1)
     .= Fifth_reduct(MS,fondamentale,
          spiral_of_fifths(MS,fondamentale,fondamentale).n1) by Def19
     .= Fifth_reduct(MS,fondamentale,r32) by Th58;
    consider r,s be positive Real such that
A2: r = r32 & s = (3 qua Real) / 2 * r & Fifth(MS,r32) = s by Th54;
A3: Fifth(MS,r32) = (9 qua Real) / 4 * @fondamentale by A2;
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 fondamentale,Octave(MS,fondamentale)
    proof
      assume
A5:   Fifth(MS,r32) is_Between fondamentale,Octave(MS,fondamentale);
A6:   ex fr be positive Real st fondamentale = fr &
        Octave(MS,fondamentale) = 2 * fr by Def15;
      thus contradiction by A5,A6,A3,XREAL_1:68;
    end;
    hence thesis by A1,A2,A4,Def18;
  end;
