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;
reserve                              MS for MusicSpace,
        fondamentale, frequency, f1, f2 for Element of MS;

theorem Th67:
  pentatonic_pythagorean_scale(MS,frequency) is pentatonic
  proof
    set scale = pentatonic_pythagorean_scale(MS,frequency);
A1: scale.1 = frequency &
    scale.2 = spiral_of_fifths(MS,frequency,frequency).2 &
    scale.3 = spiral_of_fifths(MS,frequency,frequency).4 &
    scale.4 = spiral_of_fifths(MS,frequency,frequency).1 &
    scale.5 = spiral_of_fifths(MS,frequency,frequency).3 &
    scale.6 = Octave(MS,frequency) by Def20;
    scale in 6-tuples_on the carrier of MS;
    then scale in {s where s is Element of (the carrier of MS)*: len s = 6}
      by FINSEQ_2:def 4;
    then consider s be Element of (the carrier of MS)* such that
A2: s = scale and
A3: len s = 6;
    dom s = Seg 6 by A3,FINSEQ_1:def 3;
    then reconsider g1 = scale.1, g2 = scale.2, g3 = scale.3,
      g4 = scale.4,g5 = scale.5, g6 = scale.6
      as Element of the carrier of MS by A2,FINSEQ_1:1,FINSEQ_2:11;
    now
      reconsider frequency = g1 as Element of MS;
      take frequency;
      reconsider r1 = @frequency, r2 = @g2, r3 = @g3,
        r4 = @g4, r5 = @g5, r6 = @g6 as positive Real;
      take r1,r2,r3,r4,r5,r6;
      thus scale.1 = frequency & scale.1 = r1 &
      scale.2 = r2 & scale.3 = r3 & scale.4 = r4 & scale.5 = r5 &
      scale.6 = r6;
      now
        thus r1 = 1 * @frequency;
        thus r2 = (9 qua Real) / 8 * @frequency by Th59,A1;
        thus r3 = (81 qua Real) / 64 * @frequency by A1,Th61;
        thus r4 = (3 qua Real) / 2 * @frequency by A1,Th58;
        thus r5 = (27 qua Real) / 16 * @frequency by A1,Th60;
        ex fr be positive Real st frequency = fr &
          Octave(MS,frequency) = 2 * fr by Def15;
        hence r6 = 2 * @frequency by A1;
      end;
      hence r1 < r2 < r3 & r3 < r4 < r5 & r5 < r6 by XREAL_1:68;
      thus scale.6 = Octave(MS,frequency) by A1;
    end;
    hence thesis;
  end;
