
theorem
  for n being Nat st n > 2 holds
    Lucas n = [\ (1/tau) * (Lucas (n+1) + 1/2) /]
  proof
    let n be Nat;
    assume A1: n > 2; then
A2: n > 1 by XXREAL_0:2;
A3: sqrt 5 > 0 by SQUARE_1:25;
    set tbn = tau_bar to_power n;
A4: (1/tau) * (Lucas (n+1) + 1/2) >= Lucas (n)
    proof
      tbn <= 1 / (2 * sqrt 5)
      proof
        per cases;
        suppose A5: n is even;
          n >= 2 + 1 by A1,NAT_1:13; then
          n = 3 or n > 3 by XXREAL_0:1; then
          n = 3 or n >= 3+1 by NAT_1:8; then
A6:       tbn <= tau_bar to_power 4 by Th11,A5,POLYFORM:6;
A7:       tau_bar to_power (3+1) = tau_bar to_power 3 * tau_bar to_power 1
          by Th2
          .= (2 - sqrt 5) * tau_bar by Lm8
          .= (2 - 2*sqrt 5 - sqrt 5 + (sqrt 5) ^2) / 2 by FIB_NUM:def 2
          .= (2 - 3 * sqrt 5 + 5) / 2 by SQUARE_1:def 2
          .= (7 - 3 * sqrt 5) / 2;
          sqrt 5 <= sqrt ((16/7) ^2) by SQUARE_1:26; then
          sqrt 5 <= 16/7 by SQUARE_1:def 2; then
          7 * sqrt 5 <= (16/7) * 7 by XREAL_1:64; then
          7 * sqrt 5 - 3 * 5 <= 16 - 3*5 by XREAL_1:9; then
          7 * sqrt 5 - 3 * (sqrt 5) ^2 <= 1 by SQUARE_1:def 2; then
          ((7-3*sqrt 5)*sqrt 5) / sqrt 5 <= 1 / sqrt 5 by A3,XREAL_1:72; then
          7 - 3 * sqrt 5 <= 1 / sqrt 5 by A3,XCMPLX_1:89; then
          (7 - 3 * sqrt 5) / 2 <= (1 / sqrt 5) / 2 by XREAL_1:72; then
          tau_bar to_power 4 <= 1 / (2 * sqrt 5) by A7,XCMPLX_1:78;
          hence thesis by A6,XXREAL_0:2;
        end;
        suppose n is odd; then
          tbn < 0 by Th7;
          hence thesis by A3;
        end;
      end; then
      tbn * sqrt 5 <= (1 / (2 * sqrt 5)) * sqrt 5 by A3,XREAL_1:64; then
      tbn * sqrt 5 <= 1/2 by A3,XCMPLX_1:92; then
      - tbn * sqrt 5 >= - 1/2 by XREAL_1:24; then
      tau_bar * tbn - tau * tbn + (1/2 + tau*tbn) >= - 1/2 + (1/2 + tau * tbn)
      by FIB_NUM:def 1,def 2,XREAL_1:6; then
      (tau_bar * tbn + 1/2) / tau >= (tbn * tau)/tau by XREAL_1:72; then
      (tau_bar * tbn + 1/2) / tau >= tbn by XCMPLX_1:89; then
      (tau_bar * tbn + 1/2) / tau + tau to_power (n+1) / tau >=
      tbn + tau to_power (n+1) / tau by XREAL_1:6; then
      (tau_bar * tbn + 1/2 + tau to_power (n+1))/tau  >=
      tbn + tau to_power (n+1) / tau by XCMPLX_1:62; then
      (tau_bar to_power 1 * tbn + 1/2 + tau to_power (n+1)) / tau >=
      tbn + tau to_power (n + 1) / tau; then
      (tau_bar to_power 1 * tbn + tau to_power (n+1) + 1/2) / tau >=
      tbn + (tau to_power n * tau to_power 1) / tau by Th2; then
      (tau_bar to_power (1+n) + tau to_power (n+1) + 1/2)/tau >=
      tbn + (tau to_power n * tau to_power 1) / tau by Th2; then
      (Lucas (n+1) + 1/2)/tau >=
      tbn + (tau to_power n * tau to_power 1) / tau by FIB_NUM3:21; then
      (Lucas (n+1)+1/2)/tau >= tbn+(tau to_power n*tau)/tau; then
      (Lucas (n+1) + 1/2)/tau >= tbn + tau to_power n by XCMPLX_1:89; then
      (Lucas (n+1) + 1/2)/tau >= Lucas n by FIB_NUM3:21;
      hence thesis by XCMPLX_1:99;
    end;
      (1/tau) * (Lucas (n+1) + 1/2) - 1 < Lucas n
      proof
        tbn > -1/2 by Th14,A2; then
        - tbn < - (-1/2) by XREAL_1:24; then
        (- tbn) * sqrt 5 < (1/2)*sqrt 5 by A3,XREAL_1:68; then
        tbn * tau_bar - tbn * tau + (tbn * tau + 1/2) <
        tau - 1/2 + (tbn * tau + 1/2) by FIB_NUM:def 1,def 2,XREAL_1:6; then
        tbn * tau_bar + 1/2 - tau < tau - 1/2 + tbn * tau + 1/2 - tau
          by XREAL_1:9; then
        (tbn * tau_bar + 1/2 - tau)/tau < (tbn * tau)/tau by XREAL_1:74; then
        (tbn * tau_bar) / tau + (1/2) / tau - tau / tau <
        (tbn * tau) / tau by XCMPLX_1:124; then
        (tbn * tau_bar) / tau + (1/2) /tau - tau/tau < tbn by XCMPLX_1:89; then
        (tbn * tau_bar) / tau + (1 / 2) / tau - 1 < tbn by XCMPLX_1:60; then
        (tbn * tau_bar) / tau + (1 / 2) / tau - 1 + tau to_power n <
        tbn + tau to_power n by XREAL_1:6; then
        (tbn * tau_bar) / tau + (1 / 2) / tau + tau to_power n * 1-1 < Lucas n
          by FIB_NUM3:21; then
        (tbn * tau_bar)/tau + (1 / 2) / tau + tau to_power n * (tau /tau) - 1 <
        Lucas n by XCMPLX_1:60; then
        (tbn * tau_bar)/tau + (1/2)/tau + (tau to_power n * tau)/tau - 1 <
        Lucas n by XCMPLX_1:74; then
        (tbn * tau_bar + 1/2 + tau to_power n * tau) / tau - 1 < Lucas n
          by XCMPLX_1:63; then
        (tbn * tau_bar to_power 1 + 1/2 + tau to_power n * tau) / tau - 1 <
        Lucas n; then
        (tbn * tau_bar to_power 1 + 1/2 + tau to_power n * tau to_power 1) /
        tau - 1 < Lucas n; then
        (tau_bar to_power (n+1) + 1/2 + tau to_power n * tau to_power 1) /
        tau - 1 < Lucas n by Th2; then
        (tau_bar to_power (n+1) + 1/2 + tau to_power (n+1)) / tau - 1 <
        Lucas n by Th2; then
        (tau_bar to_power (n+1) + tau to_power (n+1) + 1/2) / tau - 1 <
        Lucas n; then
        (Lucas (n+1) + 1/2) / tau - 1 < Lucas n by FIB_NUM3:21;
        hence thesis by XCMPLX_1:99;
      end;
      hence thesis by A4,INT_1:def 6;
    end;
