
theorem
  for n being Nat st n >= 2 holds
  Lucas (n) = [/ tau to_power n - 1/2 \]
  proof
    let n be Nat;
    assume A1: n >= 2; then
    n > 1 by XXREAL_0:2; then
    -1/2 <= tau_bar to_power n by Th14; then
    -1/2+tau to_power n <= tau_bar to_power n+tau to_power n by XREAL_1:6; then
A2: tau to_power n - 1/2 <= Lucas (n) by FIB_NUM3:21;
    tau_bar to_power n < 1/2 by Th8,A1; then
    tau_bar to_power n+tau to_power n < 1/2 + tau to_power n by XREAL_1:6;then
    tau to_power n - 1/2 + 1 > Lucas n by FIB_NUM3:21;
    hence thesis by A2,INT_1:def 7;
  end;
