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

theorem Th53:
  for MS1,MS2 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 frequency1 being Element of MS1
  for frequency2 being Element of MS2 st frequency1 = frequency2
  holds Octave_descendent(MS1,frequency1) = Octave_descendent(MS2,frequency2)
  proof
    let MS1,MS2 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 frequency1 be Element of MS1;
    let frequency2 be Element of MS2;
    assume
A1: frequency1 = frequency2;
    consider r1 be positive Real such that
A2: frequency1 = r1 & Octave_descendent(MS1,frequency1) = r1 / 2 by Th51;
    consider r2 be positive Real such that
A3: frequency2 = r2 & Octave_descendent(MS2,frequency2) = r2 / 2 by Th51;
    thus thesis by A1,A2,A3;
  end;
