theorem Th91:
  heptatonic_pythagorean_scale_ascending(HPS,frequency).1 = 2 * @frequency &
  heptatonic_pythagorean_scale_ascending(HPS,frequency).2
    = (9 qua Real) / 4 * @frequency &
  heptatonic_pythagorean_scale_ascending(HPS,frequency).3
    = (81 qua Real) / 32 * @frequency &
  heptatonic_pythagorean_scale_ascending(HPS,frequency).4
    = (8 qua Real) / 3 * @frequency &
  heptatonic_pythagorean_scale_ascending(HPS,frequency).5
    = (3 qua Real) * @frequency &
  heptatonic_pythagorean_scale_ascending(HPS,frequency).6
    = (27 qua Real) / 8 * @frequency &
  heptatonic_pythagorean_scale_ascending(HPS,frequency).7
    = (243 qua Real) / 64 * @frequency &
  heptatonic_pythagorean_scale_ascending(HPS,frequency).8 = 4 * @frequency
  proof
    set f2 = Octave(HPS,frequency);
    consider fr be positive Real such that
A1: frequency = fr & Octave(HPS,frequency) = 2 * fr by Def15;
    heptatonic_pythagorean_scale(HPS,f2).1 = 1 * @f2 &
    heptatonic_pythagorean_scale(HPS,f2).2
      = (9 qua Real) / 8 * @f2 &
    heptatonic_pythagorean_scale(HPS,f2).3
      = (81 qua Real) / 64 * @f2 &
    heptatonic_pythagorean_scale(HPS,f2).4
      = (4 qua Real) / 3 * @f2 &
    heptatonic_pythagorean_scale(HPS,f2).5
      = (3 qua Real) / 2 * @f2 &
    heptatonic_pythagorean_scale(HPS,f2).6
      = (27 qua Real) / 16 * @f2 &
    heptatonic_pythagorean_scale(HPS,f2).7
      = (243 qua Real) / 128 * @f2 &
    heptatonic_pythagorean_scale(HPS,f2).8 = 2 * @f2
      by Th88;
    hence thesis by A1;
  end;
