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 Th71:
  for r1,r2 being positive Real st f1 = r1 & f2 = r2 &
  r2 = (4 qua Real) / 3 * r1 holds
  (Fifth(MS,f2) is_Between fondamentale,Octave(MS,fondamentale) implies
  Octave_descendent(MS,Fifth_reduct(MS,fondamentale,f2)) = f1) &
  (not Fifth(MS,f2) is_Between fondamentale,Octave(MS,fondamentale) implies
  Fifth_reduct(MS,fondamentale,f2) = f1)
  proof
    let r1,r2 be positive Real;
    assume that
A1: f1 = r1 & f2 = r2 and
A2: r2 = (4 qua Real) / 3 * r1;
    thus Fifth(MS,f2) is_Between fondamentale,Octave(MS,fondamentale) implies
      Octave_descendent(MS,Fifth_reduct(MS,fondamentale,f2)) = f1
    proof
      assume
A3:   Fifth(MS,f2) is_Between fondamentale,Octave(MS,fondamentale);
A4:   ex fr be positive Real st f2 = fr &
        Fifth(MS,f2) = (3 qua Real) / 2 * fr by Def12;
      ex r being positive Real st Fifth_reduct(MS,fondamentale,f2) = r &
        Octave_descendent(MS,Fifth_reduct(MS,fondamentale,f2)) = r / 2
        by Th51;
      then Octave_descendent(MS,Fifth_reduct(MS,fondamentale,f2))
        = ((2 qua Real) * r1) / 2 by A3,A4,Def18,A1,A2
       .= f1 by A1;
      hence thesis;
    end;
    thus not Fifth(MS,f2) is_Between fondamentale,Octave(MS,fondamentale)
      implies Fifth_reduct(MS,fondamentale,f2) = f1
    proof
      assume
A5:   not Fifth(MS,f2) is_Between fondamentale,Octave(MS,fondamentale);
      consider fr be positive Real such that
A6:   f2 = fr & Fifth(MS,f2) = (3 qua Real) / 2 * fr by Def12;
      ex r being positive Real st Fifth(MS,f2) = r &
        Octave_descendent(MS,Fifth(MS,f2)) = r / 2 by Th51;
      hence thesis by A1,A2,A6,A5,Def18;
    end;
  end;
