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
  ex r1,r2 being Element of REALPLUS st
  intrval(f1,f2) = REAL_ratio(r1,r2)
  proof
    the carrier of MS c= REALPLUS by Def07a;
    then reconsider r1 = f1,r2 = f2 as Element of REALPLUS;
    consider r,s be positive Real such that
A1: r1 = r & r2 = s & REAL_ratio(r1,r2) = s / r by Def01;
    intrval(f1,f2) = s / r & REAL_ratio(r1,r2) = s / r by A1,Def21;
    hence thesis;
  end;
