
theorem
  for n being Nat st n >= 2 holds
  Lucas (2*n+1) = [\ tau to_power (2*n+1) /]
  proof
    let n be Nat;
    assume n >= 2; then
    2 * n >= 2 * 2 by XREAL_1:64; then
    2 * n + 1 >= 4 + 1 by XREAL_1:6; then
A1: 2 * n + 1 > 1 by XXREAL_0:2;
    tau_bar to_power (2*n+1) <= 0 by Th7; then
    tau to_power (2*n+1) + 0 >= tau to_power (2*n+1) + tau_bar to_power (2*n+1)
      by XREAL_1:6; then
A2: tau to_power (2*n+1) >= Lucas (2*n+1) by FIB_NUM3:21;
    -1/2 <= tau_bar to_power (2 * n +1 ) by Th14,A1; then
    tau_bar to_power (2 * n + 1) > -1 by XXREAL_0:2; then
    -1 + tau to_power (2*n+1) < tau to_power (2*n+1) + tau_bar to_power (2*n+1)
      by XREAL_1:6; then
    tau to_power (2*n+1) - 1 < Lucas (2*n+1) by FIB_NUM3:21;
    hence thesis by A2,INT_1:def 6;
  end;
