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 Th90:
  intrval(hepta_fondamental(HPS,frequency),hepta_1(HPS,frequency))
    = pythagorean_tone &
  intrval(hepta_1(HPS,frequency),hepta_2(HPS,frequency))
    = pythagorean_tone &
  intrval(hepta_2(HPS,frequency),hepta_3(HPS,frequency))
    = heptatonic_pythagorean_semitone &
  intrval(hepta_3(HPS,frequency),hepta_4(HPS,frequency))
    = pythagorean_tone &
  intrval(hepta_4(HPS,frequency),hepta_5(HPS,frequency))
    = pythagorean_tone &
  intrval(hepta_5(HPS,frequency),hepta_6(HPS,frequency))
    = pythagorean_tone &
  intrval(hepta_6(HPS,frequency),hepta_octave(HPS,frequency))
    = heptatonic_pythagorean_semitone
  proof
    set MS = HPS;
A1: ex fr be positive Real st frequency = fr &
      Octave(MS,frequency) = 2 * fr by Def15;
A2: heptatonic_pythagorean_scale(MS,frequency).8 = 2 * @frequency by Th88;
    ex r1,r2,r3,r4,r5,r6,r7,r8 be positive Real st
    heptatonic_pythagorean_scale(MS,frequency).1 = r1 &
    heptatonic_pythagorean_scale(MS,frequency).2 = r2 &
    heptatonic_pythagorean_scale(MS,frequency).3 = r3 &
    heptatonic_pythagorean_scale(MS,frequency).4 = r4 &
    heptatonic_pythagorean_scale(MS,frequency).5 = r5 &
    heptatonic_pythagorean_scale(MS,frequency).6 = r6 &
    heptatonic_pythagorean_scale(MS,frequency).7 = r7 &
    heptatonic_pythagorean_scale(MS,frequency).8 = r8 &
    r2 / r1 = (9 qua Real) / 8 & r3 / r2 = (9 qua Real) / 8 &
    r4 / r3 = (256 qua Real) / 243 & r5 / r4 = (9 qua Real) / 8 &
    r6 / r5 = (9 qua Real) / 8 & r7 / r6 = (9 qua Real) / 8 &
    r8 / r7 = (256 qua Real) / 243 by Lem90;
    hence thesis by A1,A2,Def21;
  end;
