
theorem Th23:
  for n being Nat st n >= 2 & n is odd holds
    Fib (n+1) = [/ tau * Fib n - 1 \]
  proof
    let n be Nat;
    assume A1: n >= 2 & n is odd;
A2: sqrt 5 > 0 by SQUARE_1:17,27;
    tau * tau_bar to_power n + sqrt 5 >= tau_bar to_power (n+1)
    proof
      set tbn = tau_bar to_power n;
A3:   tbn < 0 by A1,Th7;
      tau + sqrt 5 / tbn <= tau_bar
      proof
        n > 1 by A1,XXREAL_0:2; then
        tbn >= - 1 / 2 by Th14; then
        tbn > -1 by XXREAL_0:2; then
        tbn + 1 >= - 1 + 1 by XREAL_1:6; then
        (tbn + 1) / tbn <= 0 / tbn by A3; then
        tbn / tbn + 1/ tbn <= 0 by XCMPLX_1:62; then
        1 + 1 / tbn <= 0 by A3,XCMPLX_1:60; then
        (1 + 1 / tbn) * sqrt 5 <= 0 * sqrt 5 by A2; then
        1 * sqrt 5 + (1 / tbn) * sqrt 5 <= 0; then
        sqrt 5 + sqrt 5 / tbn <= 0 by XCMPLX_1:74; then
        sqrt 5 / 2 + sqrt 5 / tbn + sqrt 5 / 2 - sqrt 5 / 2 <= 0 - sqrt 5 / 2
        by XREAL_1:9; then
        sqrt 5 / 2 + sqrt 5 / tbn + 1/2 <= - sqrt 5 / 2 + 1/2 by XREAL_1:6;
        hence thesis by FIB_NUM:def 1,def 2;
      end; then
      (tau + sqrt 5 / tbn) * tbn >= tau_bar * tbn by A3,XREAL_1:65; then
      (tau + sqrt 5 / tbn) * tbn >= tau_bar to_power 1 * tbn; then
      tau *tbn + (sqrt 5 / tbn) * tbn >= tau_bar to_power (n+1) by Th2;
      hence thesis by A3,XCMPLX_1:87;
    end; then
    - (tau * tau_bar to_power n + sqrt 5) <= - tau_bar to_power (n+1)
    by XREAL_1:24; then
    - tau * tau_bar to_power n - sqrt 5 + tau to_power (n+1) <=
    - tau_bar to_power (n+1) +  tau to_power (n+1) by XREAL_1:6; then
    tau to_power (n+1) - tau * tau_bar to_power n - sqrt 5 <=
    tau to_power (n+1) - tau_bar to_power (n+1); then
    tau to_power 1 * tau to_power n - tau * tau_bar to_power n - sqrt 5 <=
    tau to_power (n+1) - tau_bar to_power (n+1) by POWER:27; then
    tau * tau to_power n - tau * tau_bar to_power n - sqrt 5 <=
    (tau to_power (n+1) - tau_bar to_power (n+1)); then
    (tau * (tau to_power n - tau_bar to_power n) - sqrt 5) / sqrt 5 <=
    (tau to_power (n+1)-tau_bar to_power (n+1))/sqrt 5 by A2,XREAL_1:72; then
    (tau * (tau to_power n - tau_bar to_power n))/sqrt 5 - sqrt 5 / sqrt 5 <=
    (tau to_power (n+1) - tau_bar to_power (n+1))/sqrt 5 by XCMPLX_1:120; then
    tau * ((tau to_power n - tau_bar to_power n)/sqrt 5) - sqrt 5 / sqrt 5 <=
    (tau to_power (n+1) - tau_bar to_power (n+1))/sqrt 5 by XCMPLX_1:74; then
    tau * ((tau to_power n - tau_bar to_power n) / sqrt 5) - 1 <=
    (tau to_power (n+1) - tau_bar to_power (n+1)) / sqrt 5
      by A2,XCMPLX_1:60; then
    tau * ((tau to_power n - tau_bar to_power n) / sqrt 5) - 1 <= Fib (n+1)
      by FIB_NUM:7; then
A4: tau * Fib n - 1 <= Fib (n+1) by FIB_NUM:7;
    tau * Fib n - 1 + 1 > Fib (n+1)
    proof
      set tn = tau to_power n;
      set tbn = tau_bar to_power n;
A5:   tau * Fib n = tau * ((tn - tbn) / sqrt 5) by FIB_NUM:7
      .= (tau * (tn - tbn)) / sqrt 5 by XCMPLX_1:74;
A6:   tbn < 0 by Th7,A1;
      tau * tbn < tau_bar to_power 1 * tbn by A6,XREAL_1:69; then
      tau * tbn < tau_bar to_power (n+1) by Th2; then
      - tau * tbn > - tau_bar to_power (n+1) by XREAL_1:24; then
      - tau * tbn + tau to_power (n+1) >
      - tau_bar to_power (n+1) + tau to_power (n+1) by XREAL_1:6; then
      tau to_power (n+1) - tau * tbn >
      tau to_power (n+1) - tau_bar to_power (n+1); then
      tau to_power 1 * tn - tau * tbn >
      tau to_power (n+1) - tau_bar to_power (n+1) by Th2; then
      tau * tn - tau * tbn > tau to_power (n+1) - tau_bar to_power (n+1); then
      (tau * (tn - tbn)) / sqrt 5 >
      (tau to_power (n+1) - tau_bar to_power (n+1)) / sqrt 5 by A2,XREAL_1:74;
      hence thesis by A5,FIB_NUM:7;
    end;
    hence thesis by A4,INT_1:def 7;
  end;
