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

theorem Th39:
  for frequency being Element of RAT_Music ex fr being positive Rational st
  frequency = fr & Fifth(RAT_Music,frequency) = (3 qua Rational) / 2 * fr
  proof
    set MS = RAT_Music;
    let frequency be Element of MS;
    reconsider fr = frequency as positive Rational by Th2;
    take fr;
    thus frequency = fr;
    consider fr9,qr9 be positive Rational such that
A1: fr9 = frequency and
A2: qr9 = (3 qua Rational) / 2 * fr9 and
A3: [fr9,qr9] in fifth(MS) by Th36;
    reconsider qr9 as Element of MS by Th2;
    qr9 = Fifth(MS,frequency) by A1,A3,Def11bis;
    hence thesis by A1,A2;
  end;
