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

theorem Th52:
  for MS1,MS2 being classical_octave satisfying_octave_constructible
  classical_fifth satisfying_fifth_constructible
  satisfying_harmonic_closed satisfying_Nat satisfying_interval
  satisfying_equiv MusicStruct
  for f1 being Element of MS1 for f2 being Element of MS2 st
  f1 = f2 holds Fifth(MS1,f1) = Fifth(MS2,f2) &
  Octave(MS1,f1) = Octave(MS2,f2)
  proof
    let MS1,MS2 be classical_octave satisfying_octave_constructible
    classical_fifth satisfying_fifth_constructible
    satisfying_harmonic_closed satisfying_Nat satisfying_interval
    satisfying_equiv MusicStruct;
    let f1 be Element of MS1;
    let f2 be Element of MS2;
    assume
A1: f1 = f2;
    consider fr1 be positive Real such that
A2: f1 = fr1 & Fifth(MS1,f1) = (3 qua Real) / 2 * fr1 by Def12;
    consider fr2 be positive Real such that
A3: f2 = fr2 & Fifth(MS2,f2) = (3 qua Real) / 2 * fr2 by Def12;
    thus Fifth(MS1,f1) = Fifth(MS2,f2) by A1,A2,A3;
    (ex fr be positive Real st f1 = fr & Octave(MS1,f1) = 2 * fr) &
      (ex fr be positive Real st f2 = fr & Octave(MS2,f2) = 2 * fr)
      by Def15;
    hence thesis by A1;
  end;
