reserve X for set;

theorem Th3:
  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) \ N.n
proof
  let S be SigmaField of X, N be sequence of S;
  reconsider S as non empty set;
  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) \ N.(c);
  reconsider A = N.0 as Element of S;
A1: for c being Nat, 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) \ N.c as Element of S;
    take B;
    thus thesis;
  end;
  consider F being sequence of S such that
A2: F.0 = A & 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) \ N.n
  proof
    let n be Nat;
    for a,b being Element of S,s being Nat st s = n & a = F.n &
    b = F.(n+1) holds b = N.(s+1) \ N.s by A2;
    hence thesis;
  end;
  hence thesis by A2;
end;
