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

theorem
  for n being non zero Element of NAT holds {Fib (n) * Fib (n+3), 2 *
  Fib (n+1) * Fib (n+2), (Fib (n+1)) ^2 + (Fib (n+2)) ^2} is Pythagorean_triple
proof
  let n be non zero Element of NAT;
  (Fib (n) * Fib (n+3)) ^2 + ((2 * Fib (n+1)) * Fib (n+2)) ^2 = (Fib (n))
  ^2 * (Fib (n+3)) ^2 + (2 * 2) * (Fib (n+1)) ^2 * (Fib (n+2)) ^2
    .= (Fib (n)) ^2 * (Fib (n+2) + Fib (n+1)) ^2 + 4 * (Fib (n+1)) ^2 * (Fib
  (n+2)) ^2 by Th25
    .= (Fib (n+2) - Fib (n+1)) ^2 * (Fib (n+2) + Fib (n+1)) ^2 + 4 * (Fib (n
  +1)) ^2 * (Fib (n+2)) ^2 by Th30
    .= ((Fib (n+1)) ^2 + (Fib (n+2)) ^2) ^2;
  hence thesis by PYTHTRIP:def 4;
end;
