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 Th63:
  @(spiral_of_fifths(MS,frequency,frequency).4) /
  @(spiral_of_fifths(MS,frequency,frequency).2)
  = (3 * 3 qua Real) / (2 * 2 * 2)
  proof
    spiral_of_fifths(MS,frequency,frequency).4
        = (81 qua Real) / 64 * @frequency &
      spiral_of_fifths(MS,frequency,frequency).2
        = (9 qua Real) / 8 * @frequency by Th59,Th61;
    then @(spiral_of_fifths(MS,frequency,frequency).4) /
      @(spiral_of_fifths(MS,frequency,frequency).2)
      = ((81 qua Real) / 64)/((9 qua Real) / 8) by XCMPLX_1:91;
    hence thesis;
  end;
