reserve a,b,n for Element of NAT;

theorem
  for n being Nat holds Lucas(n+1) = Fib(n) + Fib(n+2)
proof
  defpred P[Nat] means Lucas($1 +1)=Fib($1) + Fib($1+2);
A1: for k being Nat st P[k] & P[k+1] holds P[k+2]
  proof
    let k be Nat;
    assume
A2: ( P[k])& P[k+1];
    Fib((k + 2) + 2) = Fib((k+2) + 1 + 1);
    then
A3: Fib((k + 2) + 2) = Fib(k+2) + Fib((k+2) + 1) by PRE_FF:1;
    Lucas((k+2)+1) = Lucas(k+1) + Lucas((k+1)+1) by Th11
      .= Fib(k) + Fib(k+1) + Fib(k+2) + Fib((k+1)+2) by A2
      .= Fib((k+1)+1) + Fib(k+2) + Fib((k+1)+2) by PRE_FF:1
      .= Fib(k+2) + Fib((k+2)+ 2) by A3;
    hence thesis;
  end;
  0+1+1=2;
  then Fib(2) = 1 by PRE_FF:1;
  then
A4: P[1] by Th14,PRE_FF:1;
  0+1+1=2;
  then
A5: P[0] by Th11,PRE_FF:1;
  thus for k being Nat holds P[k] from FIB_NUM:sch 1 (A5, A4, A1);
end;
