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

theorem Th56BIS:
  for MS being satisfying_octave_descendent_constructible
  satisfying_octave_constructible satisfying_fifth_constructible
  satisfying_harmonic_closed satisfying_Nat satisfying_commutativity
  satisfying_interval satisfying_equiv non empty MusicStruct
  for fondamentale,frequency being Element of MS
  holds (ex seq being sequence of MS st
  (seq.0 = frequency & for n being Nat holds
  seq.(n+1) = Fifth_reduct (MS,fondamentale,seq.(n))))
  proof
    let MS be satisfying_octave_descendent_constructible
    satisfying_octave_constructible satisfying_fifth_constructible
    satisfying_harmonic_closed satisfying_Nat satisfying_commutativity
    satisfying_interval satisfying_equiv non empty MusicStruct;
    let fondamentale,frequency be Element of MS;
    defpred P[set,set,set] means
    ex x,y be Element of MS st
    x = $2 & y = $3 & y = Fifth_reduct(MS,fondamentale,x);
A1: for n be Nat for x being Element of MS ex y being Element of MS
    st P[n,x,y]
    proof
      let n be Nat;
      let x be Element of MS;
      reconsider y = Fifth_reduct(MS,fondamentale,x) as Element of MS;
      take y;
      thus thesis;
    end;
    consider seq be sequence of MS such that
A2: seq.0 = frequency and
A3: for n be Nat holds P[n,seq.n,seq.(n+1)] from RECDEF_1:sch 2(A1);
    reconsider seq as sequence of MS;
    now
      thus seq.0 = frequency by A2;
      hereby
        let n be Nat;
        consider x,y be Element of MS such that
A4:     x = seq.n and
A5:     y = seq.(n+1) and
A6:     y = Fifth_reduct(MS,fondamentale,x) by A3;
        reconsider m = n+1 as non zero Nat;
        reconsider m9 = m - 1 as Nat;
        thus seq.(n+1)=Fifth_reduct(MS,fondamentale,seq.n) by A4,A5,A6;
      end;
    end;
    hence thesis;
  end;
