
theorem
  for n being Nat st n >= 2 holds
  Fib n = [\ (1/tau) * (Fib (n+1) + 1/2)/]
  proof
    let n be Nat;
    assume A1: n >= 2; then
A2: n > 1 by XXREAL_0:2;
    set tbn = tau_bar to_power n;
    sqrt 5 < sqrt (5 ^2) by SQUARE_1:27; then
    sqrt 5 < 5 by SQUARE_1:def 2; then
A3: sqrt 5 - 5 < 5 - 5 by XREAL_1:9;
A4: (1 / tau) * (Fib (n+1) + 1/2) >= Fib n
    proof
      set tbm = tau_bar to_power (n+1);
      set tn = tau to_power n;
      tbm / tau - sqrt 5 / (2 * tau) <= tbn
      proof
        per cases;
        suppose A5: n is even;
          sqrt 5 / ((sqrt 5 - 5) * tau) <= tbn by Lm21,A2,Th14; then
          (sqrt 5 / ((sqrt 5 - 5) * tau)) *  ((sqrt 5 - 5) * tau) >=
          tbn * ((sqrt 5 - 5) * tau) by A3,XREAL_1:65; then
A6:       sqrt 5 >= tbn * ((sqrt 5 - 5) * tau) by A3,XCMPLX_1:87;
A7:       tbn > 0 by Th6,A5; then
          sqrt 5 / tbn >= (tau *(sqrt 5 - 5)*tbn) / tbn by A6,XREAL_1:72; then
          sqrt 5 / tbn >= tau * (sqrt 5 - 5) by A7,XCMPLX_1:89; then
          (sqrt 5/tbn) / tau >= ((sqrt 5 - 5)*tau) / tau by XREAL_1:72; then
          (sqrt 5 / tbn) / tau >= sqrt 5 - 5 by XCMPLX_1:89; then
          sqrt 5 / (tbn * tau) >= sqrt 5 - 5 by XCMPLX_1:78; then
          sqrt 5 /(tbn*tau)+(-sqrt 5+5)>=sqrt 5-5+(-sqrt 5+5) by XREAL_1:6;then
          - (sqrt 5 / (tbn * tau) - sqrt 5 + 5) <= 0; then
          - (sqrt 5 / (tbn*tau)) + sqrt 5 - 5 + 2 <= 0 + 2 by XREAL_1:6; then
          (- sqrt 5/(tbn*tau) + (sqrt 5 - 3)) / 2 <= 2 / 2 by XREAL_1:72; then
          - sqrt 5 / (tbn * tau) / 2 + tau_bar / tau <= 1 by Lm4; then
          - sqrt 5 / (tbn * tau * 2) + tau_bar / tau <= 1 by XCMPLX_1:78; then
          (tau_bar/tau-sqrt 5/(2*tbn*tau))*tbn <= 1*tbn by A7,XREAL_1:64; then
          (tau_bar/tau) * tbn - (sqrt 5/(2*tbn*tau)) * tbn <= tbn; then
          (tau_bar*tbn)/tau-(sqrt 5/(2*tbn*tau))*tbn <= tbn by XCMPLX_1:74;then
          (tau_bar to_power 1 * tbn) / tau - (sqrt 5/(2*tbn*tau)) * tbn <= tbn;
then
          tau_bar to_power (n+1) / tau - (sqrt 5 / ((2*tau)*tbn)) * tbn <= tbn
            by Th2; then
          tau_bar to_power (n+1) / tau - ((sqrt 5/(2*tau))/tbn) * tbn <= tbn
            by XCMPLX_1:78;
          hence thesis by A7,XCMPLX_1:87;
        end;
        suppose n is odd; then
A8:       tbn < 0 by Th7;
          sqrt 5 / (tau * (sqrt 5-5)) <= tbn by Lm21,A2,Th14;  then
          (sqrt 5 / (tau * (sqrt 5-5))) * (sqrt 5 - 5) >= tbn * (sqrt 5 - 5)
            by A3,XREAL_1:65; then
          (sqrt 5 / tau / (sqrt 5-5)) * (sqrt 5 - 5) >= tbn * (sqrt 5 - 5)
            by XCMPLX_1:78; then
          (sqrt 5/ tau) >= tbn * (sqrt 5 - 5) by A3,XCMPLX_1:87; then
          (sqrt 5/ tau)/tbn <= (tbn * (sqrt 5 - 5))/tbn by A8,XREAL_1:73; then
          sqrt 5 / (tau * tbn) <= (tbn * (sqrt 5 - 5))/tbn by XCMPLX_1:78; then
          sqrt 5 / (tau * tbn) <= sqrt 5 - 5 by A8,XCMPLX_1:89; then
          - (sqrt 5 / (tau * tbn)) >= -(sqrt 5 - 5) by XREAL_1:24; then
          - (sqrt 5 / (tau * tbn)) + (sqrt 5-3)>= - sqrt 5 + 5 + (sqrt 5-3)
            by XREAL_1:6; then
          (-(sqrt 5 / (tau * tbn)) + (sqrt 5-3)) /2 >= 2 /2 by XREAL_1:72; then
          tau_bar /tau - (sqrt 5 / (tau * tbn)) / 2 >= 1 by Lm4; then
          tau_bar /tau - (sqrt 5 / (tau * tbn*2)) >= 1 by XCMPLX_1:78; then
          (tau_bar/tau-sqrt 5/(tau*tbn*2))*tbn <= 1*tbn by A8,XREAL_1:65; then
          (tau_bar/tau)*tbn - (sqrt 5/(tau*2*tbn))*tbn <= tbn; then
          (tau_bar/tau)*tbn-(sqrt 5/(tau*2)/tbn)*tbn <= tbn by XCMPLX_1:78;then
          (tau_bar /tau)*tbn - sqrt 5 / (tau*2) <= tbn by A8,XCMPLX_1:87; then
          tau_bar * tbn/tau - sqrt 5 / (tau * 2) <= tbn by XCMPLX_1:74; then
          tau_bar to_power 1 * tbn/tau - sqrt 5 / (tau * 2) <= tbn;
          hence thesis by Th2;
        end;
      end; then
      - (tbm / tau - (sqrt 5 / (2 * tau))) >= - tbn by XREAL_1:24; then
      - tbm/tau + sqrt 5/(2*tau) + (tn*tau)/tau >= - tbn + (tn*tau)/tau
      by XREAL_1:6; then
      (tn*tau)/tau - tbm/tau + sqrt 5/(2*tau) >= -tbn + (tn*tau)/tau; then
      (tn*tau-tbm)/tau+sqrt 5/(2*tau) >= -tbn+(tn*tau)/tau by XCMPLX_1:120;then
      (tn*tau - tbm)/tau + sqrt 5/(2*tau) >= -tbn + tn by XCMPLX_1:89; then
      (tn*tau to_power 1-tbm)/tau + sqrt 5/(2*tau) >= tn-tbn; then
      (tau to_power (n+1) - tbm) / tau + sqrt 5 / (2 * tau) >= tn - tbn
        by Th2; then
      ((tau to_power (n+1) - tbm) / tau + sqrt 5 / (2 * tau)) / sqrt 5 >=
      (tn - tbn) / sqrt 5 by Lm20,XREAL_1:72; then
      ((tau to_power (n+1) - tbm) /tau + sqrt 5 /(2*tau)) /sqrt 5 >= Fib n
        by FIB_NUM:7; then
      ((tau to_power (n+1) - tbm) /tau) /sqrt 5 + (sqrt 5 /(2*tau)) /sqrt 5 >=
      Fib n by XCMPLX_1:62; then
      ((tau to_power (n+1) - tbm)/sqrt 5) /tau + (sqrt 5/(2*tau)) /sqrt 5 >=
      Fib n by XCMPLX_1:48; then
      Fib (n+1)/tau + (sqrt 5/(2*tau))/sqrt 5 >= Fib n by FIB_NUM:7; then
      Fib (n+1)/tau + 1/(2*tau)*(sqrt 5/sqrt 5) >= Fib n by XCMPLX_1:104; then
      Fib (n+1)/tau + 1/(2*tau) * 1 >= Fib n by Lm20,XCMPLX_1:60; then
      Fib (n+1) / tau + (1/2) / tau >= Fib n by XCMPLX_1:78; then
      (Fib (n+1) + 1/2) / tau >= Fib n by XCMPLX_1:62;
      hence thesis by XCMPLX_1:99;
    end;
    (1/tau) * (Fib (n+1) + 1/2) - 1 < Fib n
    proof
      1 < sqrt 5 by SQUARE_1:18,27; then
      1 / 2 <  sqrt 5 / 2 by XREAL_1:74; then
      tbn < sqrt 5 / 2 by Th8,A1,XXREAL_0:2; then
      tbn * sqrt 5 < (tau - 1/2) * sqrt 5 by Lm20,FIB_NUM:def 1,XREAL_1:68;
 then
      tbn * tau - tbn * tau_bar < tau * sqrt 5 - ((1*sqrt 5) / 2)
      by FIB_NUM:def 1,def 2; then
      tbn*tau-tbn*tau_bar to_power 1 < tau*sqrt 5-(sqrt 5/2); then
      tbn*tau-tau_bar to_power (n+1) < tau*sqrt 5-(sqrt 5/2) by Th2;then
      tbn * tau - tau_bar to_power (n+1) + sqrt 5 /2 <
      tau * sqrt 5 - (sqrt 5/2) + sqrt 5/2 by XREAL_1:6; then
      tbn * tau - tau_bar to_power (n+1) + sqrt 5 / 2 - tbn * tau <
      tau * sqrt 5 - tbn * tau by XREAL_1:9; then
      - tau_bar to_power (n+1) + sqrt 5 /2 - tau * sqrt 5 <
      tau * sqrt 5 - tbn * tau - tau * sqrt 5 by XREAL_1:9; then
      (- tau_bar to_power (n+1) + (sqrt 5 / 2) - (tau * sqrt 5)) / tau <
      (- tbn * tau) / tau by XREAL_1:74; then
      (-tau_bar to_power (n+1))/tau + (sqrt 5/2)/tau - (tau*sqrt 5)/tau <
      (-tbn * tau) / tau by XCMPLX_1:124; then
      -tau_bar to_power (n+1)/tau + (sqrt 5/2)/tau - tau*sqrt 5/tau <
      ((-tbn) * tau) / tau by XCMPLX_1:187; then
      - tau_bar to_power (n+1) /tau + (sqrt 5/2) /tau - tau*sqrt 5 /tau < -tbn
      by XCMPLX_1:89; then
      - tau_bar to_power (n+1) /tau + (sqrt 5/2) /tau - sqrt 5 < -tbn
      by XCMPLX_1:89; then
      - tau_bar to_power (n+1)/tau+(sqrt 5/2)/tau- sqrt 5+ tau to_power n * 1 <
      - tbn + tau to_power n by XREAL_1:6; then
      (-tau_bar to_power (n+1)/tau + (sqrt 5/2)/tau-sqrt 5+tau to_power n*1)/
      sqrt 5 < (tau to_power n - tbn)/sqrt 5 by Lm20,XREAL_1:74; then
      ( -tau_bar to_power (n+1)/tau + (sqrt 5/2)/tau - sqrt 5 +
      tau to_power n * 1)/sqrt 5 < Fib n by FIB_NUM:7; then
      ( -tau_bar to_power (n+1)/tau + (sqrt 5/2)/tau - sqrt 5 +
      tau to_power n * (tau/tau))/sqrt 5 < Fib n by XCMPLX_1:60; then
      ( -tau_bar to_power (n+1)/tau + (sqrt 5/2)/tau +
      tau to_power n * (tau/tau) - sqrt 5)/sqrt 5 < Fib n; then
      ( -tau_bar to_power (n+1)/tau + (sqrt 5/2)/tau +
      (tau to_power n * tau)/tau - sqrt 5)/sqrt 5 < Fib n
      by XCMPLX_1:74; then
      ( -tau_bar to_power (n+1)/tau + (sqrt 5/2)/tau +
      (tau to_power n * tau)/tau)/sqrt 5 - sqrt 5/sqrt 5 < Fib n
      by XCMPLX_1:120; then
      ( -tau_bar to_power (n+1)/tau + (sqrt 5/2)/tau +
      (tau to_power n * tau)/tau)/sqrt 5 - 1 < Fib n by Lm20,XCMPLX_1:60; then
      (( -tau_bar to_power (n+1))/tau + (sqrt 5/2)/tau +
      (tau to_power n * tau)/tau)/sqrt 5 - 1 < Fib n by XCMPLX_1:187; then
      ((-tau_bar to_power (n+1) + (sqrt 5/2) +
      (tau to_power n * tau))/tau)/sqrt 5 - 1 < Fib n by XCMPLX_1:63; then
      ((-tau_bar to_power (n+1) + (sqrt 5/2) +
      (tau to_power n * tau))/sqrt 5)/tau - 1 < Fib n by XCMPLX_1:48; then
      ((-tau_bar to_power (n+1) + (tau to_power n * tau) +
      sqrt 5/2)/sqrt 5)*(1/tau) - 1 < Fib n by XCMPLX_1:99; then
      ((-tau_bar to_power (n+1) + (tau to_power n * tau))/sqrt 5 +
      (sqrt 5/2)/sqrt 5)*(1/tau) - 1 < Fib n by XCMPLX_1:62; then
      ((-tau_bar to_power (n+1) + (tau to_power n * tau to_power 1))/sqrt 5 +
      (sqrt 5/2)/sqrt 5)*(1/tau) - 1 < Fib n; then
      (( -tau_bar to_power (n+1)+ tau to_power (n+1))/sqrt 5 +
      (sqrt 5/2)/sqrt 5)*(1/tau) - 1 < Fib n by Th2; then
      ((tau to_power (n+1) -tau_bar to_power (n+1))/sqrt 5 +
      (sqrt 5/2)/sqrt 5)*(1/tau) - 1 < Fib n; then
      (Fib (n+1) + (sqrt 5/2)/sqrt 5)*(1/tau) - 1 < Fib n by FIB_NUM:7; then
      (Fib (n+1) + (sqrt 5/sqrt 5)/2)*(1/tau) - 1 < Fib n by XCMPLX_1:48;
      hence thesis by Lm20,XCMPLX_1:60;
    end;
    hence thesis by A4,INT_1:def 6;
  end;
