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

theorem Th45:
  REAL_Music is classical_octave
  proof
    set MS = REAL_Music;
    for frequency being Element of MS
    ex fr be positive Real st
    frequency = fr & Octave(MS,frequency) = 2 * fr
    proof
      let frequency be Element of MS;
      consider fr,qr be positive Real such that
A1: fr = frequency and
A2: qr = 2 * fr and
A3: [fr,qr] in octave(MS) by Th41;
    reconsider qr as Element of MS by Th1;
      [frequency,qr] in octave(MS) by A1,A3;
      then Octave(MS,frequency) = 2 * fr by Def14,A2;
      hence thesis by A1;
    end;
    hence thesis;
  end;
