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;
reserve       HPS for Heptatonic_Pythagorean_Score,
        frequency for Element of HPS;

theorem Th88:
  heptatonic_pythagorean_scale(HPS,frequency).1 = 1 * @frequency &
  heptatonic_pythagorean_scale(HPS,frequency).2
    = (9 qua Real) / 8 * @frequency &
  heptatonic_pythagorean_scale(HPS,frequency).3
    = (81 qua Real) / 64 * @frequency &
  heptatonic_pythagorean_scale(HPS,frequency).4
    = (4 qua Real) / 3 * @frequency &
  heptatonic_pythagorean_scale(HPS,frequency).5
    = (3 qua Real) / 2 * @frequency &
  heptatonic_pythagorean_scale(HPS,frequency).6
    = (27 qua Real) / 16 * @frequency &
  heptatonic_pythagorean_scale(HPS,frequency).7
    = (243 qua Real) / 128 * @frequency &
  heptatonic_pythagorean_scale(HPS,frequency).8 = 2 * @frequency
  proof
    set MS = HPS;
    set gamme = heptatonic_pythagorean_scale(MS,frequency);
A1: gamme.1 = spiral_of_fifths(MS,frequency,Fourth(MS,frequency)).1 &
      gamme.2 = spiral_of_fifths(MS,frequency,Fourth(MS,frequency)).3 &
      gamme.3 = spiral_of_fifths(MS,frequency,Fourth(MS,frequency)).5 &
      gamme.4 = Fourth(MS,frequency) &
      gamme.5 = spiral_of_fifths(MS,frequency,Fourth(MS,frequency)).2 &
      gamme.6 = spiral_of_fifths(MS,frequency,Fourth(MS,frequency)).4 &
      gamme.7 = spiral_of_fifths(MS,frequency,Fourth(MS,frequency)).6 &
      gamme.8
        = Octave(MS,spiral_of_fifths(MS,frequency,Fourth(MS,frequency)).1)
      by Def26;
    gamme in 8-tuples_on the carrier of MS;
    then gamme in {s where s is Element of (the carrier of MS)*: len s = 8}
      by FINSEQ_2:def 4;
    then consider s be Element of (the carrier of MS)* such that
A2: s = gamme and
A3: len s = 8;
    dom s = Seg 8 by A3,FINSEQ_1:def 3;
    then reconsider g1 = gamme.4, g2 = gamme.1, g3 = gamme.5,
    g4 = gamme.2, g5 = gamme.6, g6 = gamme.3, g7 = gamme.7,
    g8 = gamme.8 as Element of the carrier of MS by A2,FINSEQ_1:1,FINSEQ_2:11;
    reconsider frequency2 = g1 as Element of MS;
    reconsider r1 = @frequency2, r2 = @g2, r3 = @g3, r4 = @g4,
      r5 = @g5, r6 = @g6, r7 = @g7, r8 =@g8 as positive Real;
A4: ex fr be positive Real st frequency = fr &
      Octave(MS,frequency) = 2 * fr by Def15;
A5: r8 = Octave(MS,spiral_of_fifths(MS,frequency,Fourth(MS,frequency)).1)
      by Def26
      .= 2 * @frequency by A4,Th82;
    ex fr be positive Real st frequency = fr &
      Fourth(MS,frequency) = (4 qua Real) / 3 * fr by Def24;
      hence thesis by A5,A1,Th82,Th84,Th86,Th83,Th85,Th87;
  end;
