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
  intrval(penta_fondamentale(MS,frequency),penta_1(MS,frequency))
    = pythagorean_tone &
  intrval(penta_1(MS,frequency),penta_2(MS,frequency))
    = pythagorean_tone &
  intrval(penta_2(MS,frequency),penta_3(MS,frequency))
    = pentatonic_pythagorean_semiditone &
  intrval(penta_3(MS,frequency),penta_4(MS,frequency))
    = pythagorean_tone &
  intrval(penta_4(MS,frequency),penta_octave(MS,frequency))
    = pentatonic_pythagorean_semiditone
  proof
    consider r1,r2 be positive Real
    such that
A1: r1 = penta_fondamentale(MS,frequency) & r2 = penta_1(MS,frequency) &
    intrval(penta_fondamentale(MS,frequency),penta_1(MS,frequency))= r2 / r1
      by Def21;
      consider r3,r4 be positive Real
      such that
A2:   r3 = penta_1(MS,frequency) & r4 = penta_2(MS,frequency) &
      intrval(penta_1(MS,frequency),penta_2(MS,frequency))= r4 / r3
        by Def21;
      consider r5,r6 be positive Real such that
A3:   r5 = penta_2(MS,frequency) & r6 = penta_3(MS,frequency) &
      intrval(penta_2(MS,frequency),penta_3(MS,frequency))= r6 / r5
        by Def21;
      consider r7,r8 be positive Real such that
A4:   r7 = penta_3(MS,frequency) & r8 = penta_4(MS,frequency) &
      intrval(penta_3(MS,frequency),penta_4(MS,frequency))= r8 / r7
        by Def21;
      consider r9,r10 be positive Real such that
A5:   r9 = penta_4(MS,frequency) & r10 = penta_octave(MS,frequency) &
      intrval(penta_4(MS,frequency),penta_octave(MS,frequency))= r10 / r9
        by Def21;
      ex s1,s2,s3,s4,s5,s6 be positive Real st
      pentatonic_pythagorean_scale(MS,frequency).1 = s1 &
      pentatonic_pythagorean_scale(MS,frequency).2 = s2 &
      pentatonic_pythagorean_scale(MS,frequency).3 = s3 &
      pentatonic_pythagorean_scale(MS,frequency).4 = s4 &
      pentatonic_pythagorean_scale(MS,frequency).5 = s5 &
      pentatonic_pythagorean_scale(MS,frequency).6 = s6 &
      s2 / s1 = (9 qua Real) / 8 & s3 / s2 = (9 qua Real) / 8 &
      s4 / s3 = (32 qua Real) / 27 & s5 / s4 = (9 qua Real) / 8 &
      s6 / s5 = (32 qua Real) / 27 by Th69;
      hence intrval(penta_fondamentale(MS,frequency),penta_1(MS,frequency))
        = pythagorean_tone &
      intrval(penta_1(MS,frequency),penta_2(MS,frequency))
        = pythagorean_tone &
      intrval(penta_2(MS,frequency),penta_3(MS,frequency))
        = pentatonic_pythagorean_semiditone &
      intrval(penta_3(MS,frequency),penta_4(MS,frequency))
        = pythagorean_tone &
      intrval(penta_4(MS,frequency),penta_octave(MS,frequency))
        = pentatonic_pythagorean_semiditone
      by A1,A2,A3,A4,A5,Def20;
    end;
