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 Th64:
  @(spiral_of_fifths(MS,frequency,frequency).1) /
  @(spiral_of_fifths(MS,frequency,frequency).4)
    = ((32 qua Real) / 27)
  proof
    spiral_of_fifths(MS,frequency,frequency).4
      = (81 qua Real) / 64 * @frequency &
      spiral_of_fifths(MS,frequency,frequency).1
      = (3 qua Real) / 2 * @frequency
      by Th58,Th61;
    then @(spiral_of_fifths(MS,frequency,frequency).1) /
      @(spiral_of_fifths(MS,frequency,frequency).4)
      = ((3 qua Real) / 2)/((81 qua Real) / 64) by XCMPLX_1:91
     .= ((32 qua Real) / 27);
    hence thesis;
  end;
