reserve a,b,n for Element of NAT;

theorem
  for n being Nat holds Lucas(n) + Fib(n) = 2 * Fib(n+1)
proof
  defpred P[Nat] means Lucas($1)+Fib($1)=2*Fib($1+1);
  let n be Nat;
A1: for k being Nat st P[k] & P[k+1] holds P[k+2]
  proof
    let k be Nat;
    assume that
A2: P[k] and
A3: P[k+1];
    Lucas(k+2)+Fib(k+2)= Lucas(k)+Lucas(k+1)+Fib(k+2) by Th12
      .= Lucas(k)+Lucas(k+1)+(Fib(k)+Fib(k+1)) by FIB_NUM2:24
      .= 2*Fib(k+1)+2*Fib(k+1+1) by A2,A3
      .=2*(Fib(k+1)+Fib(k+2))
      .=2*Fib(k+3) by FIB_NUM2:25
      .=2*Fib((k+2)+1);
    hence thesis;
  end;
  0+1+1=2;
  then Fib(2) = 1 by PRE_FF:1;
  then
A4: P[1] by Th11,PRE_FF:1;
A5: P[0] by Th11,PRE_FF:1;
  for k being Nat holds P[k] from FIB_NUM:sch 1 (A5, A4, A1);
  hence thesis;
end;
