
theorem
  for n being Nat st n >= 2 holds
  Lucas (2*n) = [/ tau to_power (2*n) \]
  proof
    let n be Nat;
    assume A1: n >= 2;
A2: tau_bar to_power (2*n) = (tau_bar to_power n) to_power 2 by NEWTON:9
    .= (tau_bar to_power n) ^2 by POWER:46;
 n - 0 is Element of NAT by NAT_1:21;
    then tau_bar to_power n <> 0 by FIB_NUM3:1; then
    tau_bar to_power (2*n) > 0 by A2,SQUARE_1:12; then
    0 + tau to_power (2*n) <= tau to_power (2*n) + tau_bar to_power (2*n)
      by XREAL_1:6; then
A3: tau to_power (2*n) <= Lucas (2*n) by FIB_NUM3:21;
    tau_bar to_power (2*n) < 1 by Th8,A1,XXREAL_0:2; then
    tau_bar to_power (2*n) + tau to_power (2*n) < 1 + tau to_power (2*n)
      by XREAL_1:6; then
    tau to_power (2*n) + 1 > Lucas (2*n) by FIB_NUM3:21;
    hence thesis by A3,INT_1:def 7;
  end;
