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_harmonic_closed
  satisfying_Nat satisfying_interval satisfying_equiv non empty MusicStruct
  for frequency being Element of MS holds (ex seq being sequence of MS st
  seq.0 = frequency & (for n being Nat holds [seq.n,seq.(n+1)] in fifth(MS)))
  proof
    let MS be satisfying_fifth_constructible satisfying_harmonic_closed
      satisfying_Nat satisfying_interval satisfying_equiv non empty
      MusicStruct;
    let frequency be Element of MS;
    defpred P[set,set,set] means
    ex x,y be positive Real st [$2,$3] in fifth(MS);
A1: for n be Nat for x be 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;
      ex q be Element of MS st [x,q] in fifth(MS) by Def11;
      hence 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);
    take seq;
    now
      thus seq.0 = frequency by A2;
      hereby
        let n be Nat;
        P[n,seq.n,seq.(n+1)] by A3;
        hence [seq.n,seq.(n+1)] in fifth(MS);
      end;
    end;
    hence thesis;
  end;
