reserve n, k, r, m, i, j for Nat;

theorem Th31:
  Fib (n) * Fib (n+2) - (Fib (n+1)) ^2 = (-1) |^ (n+1)
proof
  defpred P[Nat] means Fib ($1) * Fib ($1+2) - (Fib ($1+1)) ^2 = (-1) |^ ($1+1
  );
A1: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
A2: Fib (k+2) - Fib (k+1) = Fib (k+1) + Fib (k) - Fib (k+1) by Th24
      .= Fib (k);
A3: Fib (k+3) - Fib (k+1) = Fib (k+2) + Fib (k+1) - Fib (k+1) by Th25
      .= Fib (k+2);
    assume P[k];
    then (-1) |^ (k+1+1) = (-1) * (Fib (k) * Fib (k+2) - (Fib (k+1)) ^2) by
NEWTON:6
      .= Fib (k+1) * Fib ((k+1)+2) - ( Fib ((k+1)+1)) ^2 by A2,A3;
    hence thesis;
  end;
A4: P[0] by PRE_FF:1;
  for n being Nat holds P[n] from NAT_1:sch 2(A4, A1);
  hence thesis;
end;
