reserve X for set;

theorem
  for S being SigmaField of X, N,F being sequence of S holds (F.0 =
N.0 & for n being Element of NAT holds F.(n+1) = N.(n+1) \ N.n & N.n c= N.(n+1)
) implies (N.0 = F.0 & for n being Element of NAT holds N.(n+1) = F.(n+1) \/ N.
  n)
proof
  let S be SigmaField of X, N,F be sequence of S;
  assume that
A1: F.0 = N.0 and
A2: for n being Element of NAT holds F.(n+1) = N.(n+1) \ N.n & N.n c= N. (n+1);
  for n being Element of NAT holds N.(n+1) = F.(n+1) \/ N.n
  proof
    let n be Element of NAT;
    F.(n+1) = N.(n+1) \ N.n by A2;
    hence thesis by A2,XBOOLE_1:45;
  end;
  hence thesis by A1;
end;
