reserve X for set;

theorem Th4:
  for S being SigmaField of X, N being sequence of S holds ex F
being sequence of S st F.0 = N.0 & for n being Nat holds F.(n+1)
  = N.(n+1) \/ F.n
proof
  let S be SigmaField of X, N be sequence of S;
  defpred P[set,set,set] means for A,B being Element of S,c being Nat
   holds (c = $1 & A = $2 & B = $3 implies B = (N.(c+1)) \/ A);
A1: for c being Nat for A being Element of S ex B being Element
  of S st P[c,A,B]
  proof
    let c be Nat, A be Element of S;
    reconsider B = N.(c+1) \/ A as Element of S;
    take B;
    thus thesis;
  end;
  consider F being sequence of S such that
A2: F.0 = N.0 & for n being Nat holds P[n,F.n,F.(n+1)] from
  RECDEF_1:sch 2(A1);
  for n being Nat holds F.(n + 1) = N.(n+1) \/ F.n by A2;
  hence thesis by A2;
end;
