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 Th66:
  @Octave(MS,frequency) /
    @(spiral_of_fifths(MS,frequency,frequency).3)
    = ((32 qua Real) / 27)
  proof
    now
      thus spiral_of_fifths(MS,frequency,frequency).3
        = (27 qua Real) / 16 * @frequency by Th60;
      ex fr be positive Real st frequency = fr &
        Octave(MS,frequency) = 2 * fr by Def15;
      hence @Octave(MS,frequency) = 2 * @frequency;
    end;
    then @Octave(MS,frequency) /
      @(spiral_of_fifths(MS,frequency,frequency).3)
        = ((4 qua Real) / 2)/((27 qua Real) / 16) by XCMPLX_1:91
       .= ((32 qua Real) / 27);
    hence thesis;
end;
