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 Th70:
  for r1,r2 being positive Real st f1 = r1 & f2 = r2 &
  r2 = (4 qua Real) / 3 * r1 holds Fifth(MS,f2) = 2 * r1 &
  not Fifth(MS,f2) is_Between f1,Octave(MS,f1)
  proof
    let r1,r2 be positive Real;
    assume that
A1: f1 = r1 and
A2: f2 = r2 and
A3: r2 = (4 qua Real) / 3 * r1;
A4: ex fr be positive Real st f2 = fr &
      Fifth(MS,f2) = (3 qua Real) / 2 * fr by Def12;
    hence Fifth(MS,f2) = 2 * r1 by A2,A3;
    ex fr be positive Real st f1 = fr & Octave(MS,f1) = 2 * fr by Def15;
    hence not Fifth(MS,f2) is_Between f1,Octave(MS,f1) by A4,A3,A2,A1;
  end;
