reserve X for set;

theorem Th13:
  for N, F being Function st (F.0 = {} & for n being Nat
   holds F.(n+1) = N.0 \ N.n & N.(n+1) c= N.n )
 for n being Nat holds F.n c= F.(n+1)
proof
  let N,F be Function;
  assume that
A1: F.0 = {} and
A2: for n being Nat holds F.(n+1) = N.0 \ N.n & N.(n+1) c= N. n;
  defpred P[Nat] means F.$1 c= F.($1+1);
A3: for n be Nat st P[n] holds P[n+1]
  proof
    let n be Nat;
    assume F.n c= F.(n+1);
    F.((n+1)+1) = N.0 \ N.(n+1) by A2;
    then N.0 \ N.n c= F.((n+1)+1) by A2,XBOOLE_1:34;
    hence thesis by A2;
  end;
  let n be Nat;
  F.(0+1) = N.0 \ N.0 by A2;
  then
A4: P[0] by A1,XBOOLE_1:37;
  for n being Nat holds P[n] from NAT_1:sch 2(A4,A3);
  hence thesis;
end;
