theorem Th69:
  ex r1,r2,r3,r4,r5,r6 being positive Real st
  pentatonic_pythagorean_scale(MS,frequency).1 = r1 &
  pentatonic_pythagorean_scale(MS,frequency).2 = r2 &
  pentatonic_pythagorean_scale(MS,frequency).3 = r3 &
  pentatonic_pythagorean_scale(MS,frequency).4 = r4 &
  pentatonic_pythagorean_scale(MS,frequency).5 = r5 &
  pentatonic_pythagorean_scale(MS,frequency).6 = r6 &
  r2 / r1 = (9 qua Real) / 8 & r3 / r2 = (9 qua Real) / 8 &
  r4 / r3 = (32 qua Real) / 27 & r5 / r4 = (9 qua Real) / 8 &
  r6 / r5 = (32 qua Real) / 27
  proof
    set r1 = penta_fondamentale(MS,frequency),
    r2 = penta_1(MS,frequency), r3 = penta_2(MS,frequency),
    r4 = penta_3(MS,frequency), r5 = penta_4(MS,frequency),
    r6 = penta_octave(MS,frequency);
    the carrier of MS c= REALPLUS by Def07a;
    then reconsider r91 = r1,r92 = r2,r93 = r3,
    r94 = r4,r95 = r5,r96 = r6 as positive Real by Th1;
A1: pentatonic_pythagorean_scale(MS,frequency).1 = r91 &
    pentatonic_pythagorean_scale(MS,frequency).2 = r92 &
    pentatonic_pythagorean_scale(MS,frequency).3 = r93 &
    pentatonic_pythagorean_scale(MS,frequency).4 = r94 &
    pentatonic_pythagorean_scale(MS,frequency).5 = r95 &
    pentatonic_pythagorean_scale(MS,frequency).6 = r96 by Def20;
    then r92 / r91 = (9 qua Real) / 8 &
    r93 / r92 = (9 qua Real) / 8 &
    r94 / r93 = (32 qua Real) / 27 &
    r95 / r94 = (9 qua Real) / 8 &
    r96 / r95 = (32 qua Real) / 27 by Th68;
    hence thesis by A1;
  end;
