reserve MS for satisfying_equiv MusicStruct;
reserve a,b,c,d,e,f for Element of MS;

theorem Th55:
  for MS being satisfying_octave_descendent_constructible
  classical_octave satisfying_octave_constructible classical_fifth
  satisfying_fifth_constructible satisfying_harmonic_closed
  satisfying_Nat satisfying_interval satisfying_equiv MusicStruct
  for fondamentale,frequency being Element of MS
  st frequency is_Between fondamentale,Octave(MS,fondamentale)
  ex r1,r2,r3 being positive Real st fondamentale = r1 & frequency = r2 &
  Octave(MS,fondamentale) = 2 * r1 & r1 <= r2 <= 2 * r1
  proof
    let MS be satisfying_octave_descendent_constructible classical_octave
    satisfying_octave_constructible classical_fifth
    satisfying_fifth_constructible satisfying_harmonic_closed satisfying_Nat
    satisfying_interval satisfying_equiv MusicStruct;
    let fondamentale,frequency be Element of MS;
    assume
A1: frequency is_Between fondamentale,Octave(MS,fondamentale);
    ex r9 be positive Real st fondamentale = r9 &
      Octave(MS,fondamentale) = 2 * r9 by Def15;
    hence thesis by A1;
  end;
