
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 1/2 >= tau_bar to_power n by Th8; then
    tau to_power n+1/2 >= tau to_power n + tau_bar to_power n by XREAL_1:6;
    then
A2: tau to_power n + 1/2 >= Lucas n by FIB_NUM3:21;
    n > 1 by A1,XXREAL_0:2; then
    1/2 - 1 < tau_bar to_power n by Th14; then
    tau to_power n+(1/2-1)< tau to_power n+tau_bar 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 6;
  end;
