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

theorem
  for MS being satisfying_fifth_constructible satisfying_harmonique_stable
  satisfying_linearite_harmonique satisfying_harmonic_closed satisfying_Nat
  satisfying_interval satisfying_equiv MusicStruct
  for frequency being Element of MS holds fifth(MS,frequency) = fifth(MS)
  proof
    let MS be satisfying_fifth_constructible satisfying_harmonique_stable
    satisfying_linearite_harmonique satisfying_harmonic_closed satisfying_Nat
    satisfying_interval satisfying_equiv MusicStruct;
    let frequency be Element of MS;
A0: now
      let x be object;
      assume
A1:   x in fifth(MS,frequency);
      then consider x1,x2 be object such that
A2:   x1 in the carrier of MS and
A3:   x2 in the carrier of MS and
A4:   x = [x1,x2] by ZFMISC_1:def 2;
      reconsider x1,x2 as Element of MS by A2,A3;
      reconsider y = 1, y1 = 2,y2 = 3 as Element of MS by Th20;
      set z = 2-harmonique(MS,frequency),
      t = 3-harmonique(MS,frequency);
      reconsider n = 2,m = 3 as non zero Nat;
      consider r1 be positive Real such that
A5:   y = r1 and
A6:   n-harmonique(MS,y) = n * r1 by Def09;
      consider r2 be positive Real such that
A7:   y = r2 and
A8:   m-harmonique(MS,y) = m * r2 by Def09;
      set a = n-harmonique(MS,frequency), b = m-harmonique(MS,frequency);
A9:   a,b equiv x1,x2 by A1,A4,EQREL_1:18;
      y1,y2 equiv a,b by A5,A6,A7,A8,Def10;
      then y1,y2 equiv x1,x2 by A9,Th23;
      hence x in fifth(MS) by A4,EQREL_1:18;
    end;
    now
      let x be object;
      assume
A11:  x in fifth(MS);
      then consider x1,x2 be object such that
A12:  x1 in the carrier of MS and
A13:  x2 in the carrier of MS and
A14:  x = [x1,x2] by ZFMISC_1:def 2;
      reconsider x1,x2 as Element of MS by A12,A13;
      reconsider y = 2,z = 3 as Element of MS by Th20;
      reconsider y9 = y as positive Real;
      reconsider n = 2,m = 3 as non zero Nat;
      set a = n-harmonique(MS,frequency), b = m-harmonique(MS,frequency),
          c = n-harmonique(MS,y), d = m-harmonique(MS,y);
A15:  a,b equiv c,d by Def10;
      reconsider n0 = 1,n1 = 2,n2 = 3 as Element of MS by Th20;
      consider r1 be positive Real such that
A16:  n0 = r1 and
A17:  n-harmonique(MS,n0) = n * r1 by Def09;
      consider r2 be positive Real such that
A18:  n0 = r2 and
A19:  m-harmonique(MS,n0) = m * r2 by Def09;
A20:  n-harmonique(MS,n0),m-harmonique(MS,n0) equiv x1,x2
        by A16,A17,A18,A19,A11,A14,EQREL_1:18;
      c,d equiv n-harmonique(MS,n0),m-harmonique(MS,n0) by Def10;
      then c,d equiv x1,x2 by A20,Th23;
      then a,b equiv x1,x2 by A15,Th23;
      hence x in fifth(MS,frequency) by A14,EQREL_1:18;
    end;
    hence thesis by A0;
  end;
