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

theorem
  for n being non zero Element of NAT holds Fib (n-'1) * Fib (n+1) - (
  Fib (n)) ^2 = (-1) |^n
proof
  let n be non zero Element of NAT;
  set a = n-'1;
A1: n >= 1 by NAT_2:19;
  then n = a + 1 by XREAL_1:235;
  then
  Fib (n-'1) * Fib (n+1) - (Fib (n)) ^2 = Fib (a) * Fib (a+2) - (Fib (a+1) ) ^2
    .= (-1) |^(n-'1+1) by Th31
    .= (-1) |^(n) by A1,XREAL_1:235;
  hence thesis;
end;
