reserve MS for satisfying_equiv MusicStruct;
reserve a,b,c,d,e,f for Element of MS;
reserve MS for 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,
  fondamentale,frequency for Element of MS;
reserve                              MS for MusicSpace,
        fondamentale, frequency, f1, f2 for Element of MS;

theorem
  for r1,r2 being positive Real st f1 = r1 & f2 = r2 &
  r2 = (4 qua Real) / 3 * r1 holds
  Fifth_reduct(MS,f1,f2) = f1
  proof
    let r1,r2 be positive Real;
    assume
A1: f1 = r1 & f2 = r2 & r2 = (4 qua Real) / 3 * r1;
    then (Fifth(MS,f2) is_Between f1,Octave(MS,f1) implies
      Octave_descendent(MS,Fifth_reduct(MS,f1,f2)) = f1) &
      (not Fifth(MS,f2) is_Between f1,Octave(MS,f1) implies
      Fifth_reduct(MS,f1,f2) = f1) by Th71;
    hence thesis by A1,Th70;
  end;
