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

theorem Th56:
  for MS 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 fondamentale,frequency being Element of MS
  st frequency is_Between fondamentale,Octave(MS,fondamentale)
  holds Fifth_reduct(MS,fondamentale,frequency) is_Between
  fondamentale,Octave(MS,fondamentale)
  proof
    let MS 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 fondamentale,frequency be Element of MS;
    assume frequency is_Between fondamentale,Octave(MS,fondamentale);
    then consider r1,r2,r3 be positive Real such that
A1: fondamentale = r1 & frequency = r2 &
      Octave(MS,fondamentale) = 2 * r1 &
    r1 <= r2 <= 2 * r1 by Th55;
    consider fr be positive Real such that
A2: frequency = fr & Fifth(MS,frequency) = (3 qua Real) / 2 * fr
      by Def12;
    per cases;
    suppose Fifth(MS,frequency) is_Between
      fondamentale,Octave(MS,fondamentale);
     hence thesis by Def18;
    end;
    suppose
A3:   not Fifth(MS,frequency) is_Between
      fondamentale,Octave(MS,fondamentale);
A4:   ex r being positive Real st Fifth(MS,frequency) = r &
      Octave_descendent(MS,Fifth(MS,frequency)) = r / 2 by Th51;
      per cases by A2,A1,A3;
      suppose
A5:     (3 qua Real) / 2 * r2 < r1;
        reconsider r32 = (3 qua Real) / 2 as non zero positive Real;
        r1 * (1 / r32) < 1 * r1 by XREAL_1:68;
        then
A6:     r1 / r32 < r1 by XCMPLX_1:99;
        r32 * r2 / r32 < r1 / r32 by A5,XREAL_1:74;
        then r2 < r1 / r32 by XCMPLX_1:89;
        hence thesis by A1,A6,XXREAL_0:2;
      end;
      suppose 2 * r1 <= (3 qua Real) / 2 * r2;
        then
A7:     2 * r1 / 2 <= (3 qua Real) / 2 * r2 / 2 by XREAL_1:72;
        reconsider r34 = (3 qua Real) / 4 as positive Real;
A8:     (3 qua Real) / 2 * r1 < 2 * r1 by XREAL_1:68;
        ((3 qua Real) / 4) * r2 <= ((3 qua Real) / 4) * (2 * r1)
          by A1,XREAL_1:66;
        then ((3 qua Real) / 4)  * r2 < 2 * r1 by A8,XXREAL_0:2;
        hence thesis by A2,A7,A1,A4,A3,Def18;
      end;
    end;
  end;
