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

theorem Th51:
  for MS being satisfying_octave_descendent_constructible
  classical_octave satisfying_octave_constructible
  satisfying_fifth_constructible satisfying_harmonic_closed
  satisfying_Nat satisfying_commutativity satisfying_interval
  satisfying_equiv satisfying_Real non empty MusicStruct
  for frequency being Element of MS
  holds ex r being positive Real st frequency = r &
  Octave_descendent(MS,frequency) = r / 2
  proof
    let MS be satisfying_octave_descendent_constructible
    classical_octave satisfying_octave_constructible
    satisfying_fifth_constructible satisfying_harmonic_closed
    satisfying_Nat satisfying_commutativity
    satisfying_interval satisfying_equiv satisfying_Real
    non empty MusicStruct;
    let frequency be Element of MS;
A1: the carrier of MS c= REALPLUS by Def07a;
    then reconsider r = frequency as positive Real by Th1;
    reconsider r2 = r/2 as positive Real;
    set ff = Octave_descendent(MS,frequency);
    reconsider rff = ff as positive Real by A1,Th1;
A2: [ff,frequency] in octave(MS) &
    [ff,Octave(MS,ff)] in octave(MS) by Def14,Def17;
    ex fr be positive Real st
    ff = fr & Octave(MS,ff) = 2 * fr by Def15;
    then frequency = 2 * rff by A2,Def14;
    then rff = r / 2;
    hence thesis;
  end;
