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
  spiral_of_fifths(MS,fondamentale,fondamentale).5
    = (243 qua Real) / 128 * @fondamentale
  proof
    reconsider n5=5,n4=4,n3=3,n2 = 2,n1 = 1,n0 = 0 as Nat;
    spiral_of_fifths(MS,fondamentale,fondamentale).n4 is Element of MS;
    then reconsider r32 = (81 qua Real) / 64 * @fondamentale as Element of MS
    by Th61;
A1: spiral_of_fifths(MS,fondamentale,fondamentale).5
      = spiral_of_fifths(MS,fondamentale,fondamentale).(n4 + 1)
     .= Fifth_reduct(MS,fondamentale,
          spiral_of_fifths(MS,fondamentale,fondamentale).n4) by Def19
     .= Fifth_reduct(MS,fondamentale,r32) by Th61;
    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 fondamentale = fr &
      Octave(MS,fondamentale) = 2 * fr by Def15;
     ((243 qua Real) / 128) * @fondamentale < 2 * @fondamentale &
       1 * @fondamentale < ((243 qua Real) / 128) * @fondamentale
       by XREAL_1:68;
    then Fifth(MS,r32) is_Between fondamentale,Octave(MS,fondamentale)
      by A2,A3;
    hence thesis by A2,A1,Def18;
  end;
