
theorem
  for n being Nat st n >= 2 & n is odd holds
  Lucas (n+1) = [\ tau * Lucas (n) + 1 /]
  proof
    let n be Nat;
    set tn = tau_bar to_power n;
    assume A1: n >= 2 & n is odd;
A2: tau * Lucas (n) + 1 >= Lucas (n+1)
    proof
      per cases by A1,XXREAL_0:1;
      suppose A3: n = 2;
        sqrt 5 >= 1 by SQUARE_1:18,26; then
        sqrt 5 * 3 >= 1 * 3 by XREAL_1:64; then
        5 + sqrt 5 * 3 >= 3 + 5 by XREAL_1:6; then
        (5 + sqrt 5 * 3) / 2 >= 8 / 2 by XREAL_1:72;
        hence thesis by A3,FIB_NUM:def 1,FIB_NUM3:14,15;
      end;
      suppose A4: n > 2;
A5:     sqrt 5 > 0 by SQUARE_1:25;
        tn>= -1/sqrt 5 & sqrt 5 > 0 by Th15,A4,SQUARE_1:25;then
        tn*(sqrt 5) >= (-1/sqrt 5)*(sqrt 5) by XREAL_1:64; then
        tn * (sqrt 5) >= sqrt 5 * ((-1)/sqrt 5) by XCMPLX_1:187; then
        tn*(sqrt 5) >=(sqrt 5*(-1))/sqrt 5 by XCMPLX_1:74; then
        tn*(sqrt 5)>= (sqrt 5/sqrt 5)*(-1) by XCMPLX_1:74; then
        tn * (sqrt 5) >= 1 * (-1) by A5,XCMPLX_1:60; then
        - (tau_bar to_power n * sqrt 5) <= -(-1) by XREAL_1:24; then
        tn * tau_bar - tn * tau <= 1 by FIB_NUM:def 1,def 2; then
        tn * tau_bar to_power 1-tn * tau <= 1; then
        tau_bar to_power (n+1)- tn*tau <= 1 by Th2; then
        tau_bar to_power (n+1) - tn * tau +
        tn*tau <= 1 + tn*tau by XREAL_1:6; then
        tn * tau + 1 + tau to_power (n+1) >=
        tau_bar to_power (n+1) + tau to_power (n+1) by XREAL_1:6; then
        tn * tau + 1 + tau to_power n * tau to_power 1 >=
        tau_bar to_power (n+1)+tau to_power (n+1) by Th2; then
        tn * tau + 1 + tau to_power n * tau >=
        tau_bar to_power (n+1) + tau to_power (n+1); then
        (tn + tau to_power n) * tau + 1 >=
        tau_bar to_power (n+1) + tau to_power (n+1); then
        Lucas (n) * tau + 1 >=
        tau_bar to_power (n+1) + tau to_power (n+1) by FIB_NUM3:21;
        hence thesis by FIB_NUM3:21;
      end;
    end;
    tau * Lucas n + 1 - 1 < Lucas (n+1)
    proof
A6:   Lucas (n+1) = tau to_power (n+1) + tau_bar to_power (n+1) by FIB_NUM3:21;
A7:   tau*Lucas n = tau * (tau to_power n + tn) by FIB_NUM3:21
      .= tau * tau to_power n + tau * tau_bar to_power n
      .= tau to_power 1 * tau to_power n + tau * tau_bar to_power n
      .= tau to_power (n+1) + tau * tau_bar to_power n by Th2;
   tn < 0 by Th7,A1;
      then tau*tn < tau_bar*tn by XREAL_1:69;then
      tau * tn < tau_bar to_power 1 * tn; then
      tau * tn < tau_bar to_power (n+1) by Th2;
      hence thesis by A6,A7,XREAL_1:6;
    end;
    hence thesis by A2,INT_1:def 6;
  end;
