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;

theorem Th57:
  frequency is_Between fondamentale,Octave(MS,fondamentale) implies
  for n being Nat holds spiral_of_fifths(MS,fondamentale,frequency).n
  is_Between fondamentale,Octave(MS,fondamentale)
  proof
    assume
A1: frequency is_Between fondamentale,Octave(MS,fondamentale);
    let n be Nat;
    defpred P[Nat] means
    spiral_of_fifths(MS,fondamentale,frequency).$1
      is_Between fondamentale,Octave(MS,fondamentale);
A2: P[0] by Def19,A1;
A3: for k be Nat st P[k] holds P[k+1]
    proof
      let k be Nat;
      assume A4: P[k];
      spiral_of_fifths(MS,fondamentale,frequency).(k+1)
        = Fifth_reduct(MS,fondamentale,
          spiral_of_fifths(MS,fondamentale,frequency).k)
        by Def19;
      hence thesis by A4,Th56;
    end;
    for k be Nat holds P[k] from NAT_1:sch 2(A2,A3);
    hence thesis;
  end;
