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 Th65:
  @(spiral_of_fifths(MS,frequency,frequency).3) /
  @(spiral_of_fifths(MS,frequency,frequency).1)
    = ((9 qua Real) / 8)
  proof
    spiral_of_fifths(MS,frequency,frequency).3
      = (27 qua Real) / 16 * @frequency &
    spiral_of_fifths(MS,frequency,frequency).1 = (3 qua Real) / 2 * @frequency
      by Th58,Th60;
    then @(spiral_of_fifths(MS,frequency,frequency).3) /
    @(spiral_of_fifths(MS,frequency,frequency).1)
      = ((27 qua Real) / 16)/((3 qua Real) / 2) by XCMPLX_1:91
     .= ((9 qua Real) / 8);
    hence thesis;
  end;
