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 Th88BIS:
  heptatonic_pythagorean_scale(HPS,frequency) is heptatonic
  proof
    set MS = HPS;
    set gamme = heptatonic_pythagorean_scale(MS,frequency);
    now
      now
        reconsider r1 = 1 * @frequency,
          r2 = (9 qua Real) / 8 * @frequency,
          r3 = (81 qua Real) / 64 * @frequency,
          r4 = (4 qua Real) / 3 * @frequency,
          r5 = (3 qua Real) / 2 * @frequency,
          r6 = (27 qua Real) / 16 * @frequency,
          r7 = (243 qua Real) / 128 * @frequency,
          r8 = 2 * @frequency as positive Real;
        take r1,r2,r3,r4,r5,r6,r7,r8;
        heptatonic_pythagorean_scale(MS,frequency).1 = 1 * @frequency &
          heptatonic_pythagorean_scale(MS,frequency).2
            = (9 qua Real) / 8 * @frequency &
          heptatonic_pythagorean_scale(MS,frequency).3
            = (81 qua Real) / 64 * @frequency &
          heptatonic_pythagorean_scale(MS,frequency).4
            = (4 qua Real) / 3 * @frequency &
          heptatonic_pythagorean_scale(MS,frequency).5
            = (3 qua Real) / 2 * @frequency &
          heptatonic_pythagorean_scale(MS,frequency).6
            = (27 qua Real) / 16 * @frequency &
          heptatonic_pythagorean_scale(MS,frequency).7
            = (243 qua Real) / 128 * @frequency &
          heptatonic_pythagorean_scale(MS,frequency).8
            = 2 * @frequency by Th88;
        hence gamme.1 = frequency & gamme.1 = r1 &
          gamme.2 = r2 & gamme.3 = r3 & gamme.4 = r4 & gamme.5 = r5 &
          gamme.6 = r6 & gamme.7 = r7 & gamme.8 = r8;
        thus r1 < r2 & r2 < r3 & r3 < r4 & r4 < r5 & r5 < r6 &
        r6 < r7 & r7 < r8 by XREAL_1:98;
      end;
      hence ex frequency be Element of MS,
        r1,r2,r3,r4,r5,r6,r7,r8 be positive Real st
        gamme.1 = r1 & gamme.2 = r2 & gamme.3 = r3 & gamme.4 = r4 &
        gamme.5 = r5 & gamme.6 = r6 & gamme.7 = r7 & gamme.8 = r8 &
        r1 < r2 < r3 & r3 < r4 < r5 & r5 < r6 & r6 < r7 & r7 < r8;
A1:     ex fr be positive Real st frequency = fr &
        Octave(MS,frequency) = 2 * fr by Def15;
        heptatonic_pythagorean_scale(MS,frequency).1 = 1 * @frequency &
        heptatonic_pythagorean_scale(MS,frequency).2
          = (9 qua Real) / 8 * @frequency &
        heptatonic_pythagorean_scale(MS,frequency).3
          = (81 qua Real) / 64 * @frequency &
        heptatonic_pythagorean_scale(MS,frequency).4
          = (4 qua Real) / 3 * @frequency &
        heptatonic_pythagorean_scale(MS,frequency).5
          = (3 qua Real) / 2 * @frequency &
        heptatonic_pythagorean_scale(MS,frequency).6
          = (27 qua Real) / 16 * @frequency &
        heptatonic_pythagorean_scale(MS,frequency).7
          = (243 qua Real) / 128 * @frequency &
        heptatonic_pythagorean_scale(MS,frequency).8
          = 2 * @frequency by Th88;
        hence gamme.8 = Octave(MS,frequency) &
        gamme.1 = frequency by A1;
    end;
    hence thesis;
  end;
