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

theorem
  for MS1,MS2 being satisfying_octave_descendent_constructible
  classical_octave satisfying_octave_constructible classical_fifth
  satisfying_fifth_constructible satisfying_harmonic_closed satisfying_Nat
  satisfying_commutativity satisfying_interval satisfying_equiv
  satisfying_Real non empty MusicStruct
  for frequency1,fondamentale1 being Element of MS1
  for frequency2,fondamentale2 being Element of MS2 st
  frequency1 = frequency2 & fondamentale1 = fondamentale2 holds
  Fifth_reduct(MS1,fondamentale1,frequency1)
   = Fifth_reduct(MS2,fondamentale2,frequency2)
  proof
    let MS1,MS2 be satisfying_octave_descendent_constructible
    classical_octave satisfying_octave_constructible
    classical_fifth satisfying_fifth_constructible
    satisfying_harmonic_closed satisfying_Nat satisfying_commutativity
    satisfying_interval satisfying_equiv satisfying_Real non empty
    MusicStruct;
    let frequency1,fondamentale1 be Element of MS1;
    let frequency2,fondamentale2 be Element of MS2;
    assume that
A1: frequency1 = frequency2 and
A2: fondamentale1 = fondamentale2;
    per cases;
    suppose
A3:   Fifth(MS1,frequency1) is_Between
       fondamentale1,Octave(MS1,fondamentale1); then
A4: Fifth_reduct(MS1,fondamentale1,frequency1) = Fifth(MS1,frequency1)
        by Def18;
      Fifth(MS1,frequency1) = Fifth(MS2,frequency2) &
        fondamentale1 = fondamentale2 &
        Octave(MS2,fondamentale2)=Octave(MS1,fondamentale1) by A1,A2,Th52;
      then Fifth(MS2,frequency2) is_Between
        fondamentale2,Octave(MS2,fondamentale2) by A3;
      then Fifth(MS2,frequency2)
        = Fifth_reduct(MS2,fondamentale2,frequency2)
        by Def18;
      hence thesis by A4,A1,Th52;
    end;
    suppose
A5:   not Fifth(MS1,frequency1) is_Between
        fondamentale1,Octave(MS1,fondamentale1);
      then
A6:   Fifth_reduct(MS1,fondamentale1,frequency1)
        = Octave_descendent(MS1,Fifth(MS1,frequency1))
        by Def18;
      Fifth(MS1,frequency1) = Fifth(MS2,frequency2) &
        fondamentale1 = fondamentale2 &
        Octave(MS2,fondamentale2)=Octave(MS1,fondamentale1) by A1,A2,Th52;
      then not Fifth(MS2,frequency2) is_Between
        fondamentale2,Octave(MS2,fondamentale2) by A5; then
A7:   Fifth_reduct(MS2,fondamentale2,frequency2)
        = Octave_descendent(MS2,Fifth(MS2,frequency2)) by Def18;
      Fifth(MS1,frequency1) = Fifth(MS2,frequency2) by A1,Th52;
      hence thesis by A6,A7,Th53;
    end;
  end;
