
theorem
  for n being Nat st n >= 4 holds
  Lucas (n+1) = [\ (Lucas n + 1 + sqrt(5*(Lucas n)^2 - 2*Lucas n + 1))/2 /]
  proof
    let n be Nat;
    set tb = tau_bar;
    set tbn = tau_bar to_power n;
    set tn = tau to_power n;
    assume A1: n >= 4;
A2: sqrt 5 > 0 by SQUARE_1:25;
A3: (Lucas n + 1 + sqrt (5 * (Lucas n) ^2 - 2 * Lucas n + 1)) /2 >= Lucas (n+1)
    proof
A4:   n - 0 is non zero Element of NAT by A1,NAT_1:21;
      Lucas (n+1) >= n + 1 & n + 1 >= 0 + 1 by FIB_NUM3:17,XREAL_1:6; then
      Lucas (n+1) >= Lucas n & Lucas (n+1) >= 1
      by A4,FIB_NUM3:18,XXREAL_0:2; then
      Lucas (n+1) + Lucas (n+1) >= Lucas n + 1 by XREAL_1:7; then
      Lucas (n+1)+Lucas (n+1)-Lucas n >= Lucas n+1-Lucas n by XREAL_1:9; then
A5:   2 * Lucas (n+1) - Lucas n - 1 >= 1 - 1 by XREAL_1:9;
      2*Lucas (n+1)-Lucas n - 1 = 2 * Lucas (n+1) - (tn+tbn) - 1 by FIB_NUM3:21
      .= 2 * (tau to_power (n+1) + tau_bar to_power (n+1)) - (tn + tbn) - 1
      by FIB_NUM3:21
      .= 2 *(tn*tau to_power 1+tau_bar to_power (n+1))-(tn+tbn)-1 by Th2
      .= 2 * (tn*tau + tau_bar to_power (n+1)) - (tn+tbn) - 1
      .= 2 * (tn*tau + tbn * tb to_power 1) - (tn+tbn) - 1 by Th2
      .= 2 * (tn * tau + tbn * tb) - (tn + tbn) - 1
      .= (tn - tbn) * sqrt 5 - 1 by FIB_NUM:def 1,def 2
      .= (((tn - tbn) / sqrt 5) * sqrt 5) * sqrt 5 - 1 by A2,XCMPLX_1:87
      .= (Fib n * sqrt 5) * sqrt 5 - 1 by FIB_NUM:7
      .= Fib n * (sqrt 5) ^2 - 1
      .= 5 * Fib n - 1 by SQUARE_1:def 2; then
A6:   (2 * Lucas (n+1) - Lucas n - 1)^2 = (5 * Fib n)^2 - 2 * (5*Fib n)*1 + 1^2
      by SQUARE_1:5
      .= 25 * (Fib n) ^2 - 10 * Fib n + 1;
      5 * (Lucas n) ^2 - 2 * Lucas n >= 25 * (Fib n) ^2 - 10 * Fib n
      proof
        per cases by A1,XXREAL_0:1;
        suppose n = 4;
          hence thesis by FIB_NUM2:23,FIB_NUM3:16;
        end;
        suppose n > 4; then
A7:       n >= 4+1 by NAT_1:13;
          set s5 = sqrt 5;
A8:       5 *(Lucas n)^2 - 2*Lucas n =5*(Lucas n)^2 - 2*(tn+tbn) by FIB_NUM3:21
          .= 5 * (tn + tbn) ^2 - 2 * (tn + tbn) by FIB_NUM3:21
          .= 5 * tn ^2 + 5 * 2 * tn * tbn + 5 * tbn ^2 - 2 * tn - 2 * tbn;
A9:       25 * (Fib n)^2 - 10 * Fib n = 25 * (Fib n)^2 - 10 * ((tn-tbn)/s5)
          by FIB_NUM:7
          .= 25 * ((tn-tbn) / s5)^2 - 10 * ((tn-tbn) / s5) by FIB_NUM:7
          .= 25 * ((tn-tbn)^2/s5^2)- 10 *((tn-tbn)/s5) by XCMPLX_1:76
          .= 25 * ((tn-tbn)^2 / 5) - 10 * ((tn-tbn)/s5) by SQUARE_1:def 2
          .= 5 *tn^2 - 5*2*tn*tbn + 5*tbn^2 - 10*(((tn-tbn)*s5)/s5^2)
          by A2,XCMPLX_1:91
          .= 5 * tn^2 - 10*tn*tbn + 5 * tbn^2 - 10 * (((tn-tbn)*s5)/5)
          by SQUARE_1:def 2
          .= 5 * tn^2 - 10*tn*tbn + 5 * tbn^2 - 2*tn*s5 + 2 * tbn * s5;
A10:       n -' 1 + 1 = n + 1 -' 1 by A1,NAT_D:38,XXREAL_0:2
          .= n by NAT_D:34;
A11:       -10 <= 10 * (-1) to_power n by Lm1;
          n -' 1 >= 5 -' 1 by A7,NAT_D:42; then
          n -' 1 >= 5 - 1 by NAT_D:39; then
          Lucas (n-'1) >= 7 by Th12,FIB_NUM3:16; then
          Lucas (n-'1) >= 5 by XXREAL_0:2; then
          Lucas (n-'1) * (-2) <= 5 * (-2) by XREAL_1:65; then
          Lucas (n-'1+1) - (2 * Lucas (n-'1) + Lucas (n-'1+1)) <= -10; then
          Lucas n - 5 * Fib n <= -10 by A10,FIB_NUM3:22; then
          tn + tbn - 5 * Fib n <= -10 by FIB_NUM3:21; then
          tn + tbn - 5 * ((tn-tbn) / s5) <= -10 by FIB_NUM:7; then
          tn + tbn - 5 * ((tn-tbn) * (1/s5)) <= -10 by XCMPLX_1:99; then
          tn + tbn - 5 * (1/s5) * (tn-tbn) <= -10; then
          tn +tbn-(s5)^2*(1/s5)*(tn-tbn) <= -10 by SQUARE_1:def 2; then
          tn + tbn - s5 * (s5 * (1/s5)) * (tn-tbn) <= -10; then
          tn+tbn-s5*(s5/s5)*(tn-tbn) <= -10 by XCMPLX_1:99; then
          tn + tbn - s5*1*(tn-tbn) <= -10 by A2,XCMPLX_1:60; then
          tn + tbn - s5 * tn + s5 * tbn <= 10 * (tau * tb) to_power n
          by Lm3,A11,XXREAL_0:2; then
          tn + tbn - s5 * tn + sqrt 5 * tbn - 10 * (tau*tb) to_power n <=
          10*(tau*tb) to_power n - 10*(tau*tb) to_power n by XREAL_1:9; then
          tn + tbn - s5 * tn + s5 * tbn - 10 * (tau*tb) to_power n +
          sqrt 5 * tn <= 0 + s5 * tn by XREAL_1:6; then
          tn + tbn + s5 * tbn - 10*(tau*tb) to_power n <= s5 * tn; then
          tn +tbn+s5*tbn - 10*(tn*tbn) <= s5*tn by NEWTON:7; then
          (-10 * tn * tbn + tn + tbn + tbn * s5) * 2 <= (tn * s5) * 2
          by XREAL_1:64; then
          -(- 20 * tn * tbn + 2 * tn + 2 * tbn + 2*tbn*s5) >= - 2*tn*s5
          by XREAL_1:24; then
          10*tn*tbn - 2*tn - 2*tbn - 2*tbn*s5 + 10*tn*tbn - 10*tn*tbn >=
          - 2 * tn * s5 - 10 * tn * tbn by XREAL_1:9; then
          10*tn*tbn - 2*tn-2*tbn - 2*tbn*s5 + 2*tbn*s5 >=
          - 10*tn*tbn - 2*tn*s5 + 2*tbn*s5 by XREAL_1:6; then
          10 * tn * tbn - 2 * tn - 2 * tbn + 5 * tbn ^2 >=
          - 10*tn*tbn - 2*tn*s5 + 2*tbn*s5 + 5*tbn^2 by XREAL_1:6; then
          10 * tn * tbn + 5 * tbn ^2 - 2 * tn - 2 * tbn + 5 * tn ^2 >=
          - 10*tn*tbn + 5*tbn^2 - 2*tn*s5 + 2 * tbn * s5 + 5 * tn ^2
          by XREAL_1:6;
          hence thesis by A8,A9;
        end;
      end; then
      5 * (Lucas n) ^2 - 2 * Lucas n + 1 >= (2 * Lucas (n+1) - Lucas n - 1) ^2
      by A6,XREAL_1:6; then
      sqrt (5*(Lucas n)^2 - 2*Lucas n + 1) >=
      sqrt ((2 * Lucas (n+1) - Lucas n - 1) ^2) by SQUARE_1:26; then
      sqrt (5 * (Lucas n) ^2 - 2 * Lucas n + 1) >= 2 *Lucas (n+1) - Lucas n - 1
      by A5,SQUARE_1:def 2; then
      sqrt (5 * (Lucas n) ^2 - 2 * Lucas n + 1)+(Lucas n + 1) >=
      2 * Lucas (n+1) - Lucas n - 1 + (Lucas n + 1) by XREAL_1:6; then
      (Lucas n + 1 + sqrt (5 * (Lucas n) ^2 - 2 * Lucas n + 1)) / 2 >=
      (2 * Lucas (n+1)) / 2 by XREAL_1:72;
      hence thesis;
    end;
    (Lucas n + 1 + sqrt (5*(Lucas n) ^2 - 2 * Lucas n + 1))/2 - 1 < Lucas (n+1)
    proof
      Lucas n >= n by FIB_NUM3:17; then
A12:   Lucas n >= 4 by A1,XXREAL_0:2; then
A13:   Lucas n / 5 >= 4 / 5 by XREAL_1:72;
      Fib n >= 3 by A1,FIB_NUM2:23,45; then
      Fib n + Lucas n / 5 >= 3 + 4/5 by A13,XREAL_1:7; then
A14:   2 < Fib n + Lucas n / 5 by XXREAL_0:2;
      n is even or n is odd; then
      (-1) to_power n <= 1 by FIB_NUM2:2,3; then
      2 * (-1) to_power n <= 2 * 1 by XREAL_1:64; then
      2 * (-1) to_power n < Fib n + Lucas n / 5 by A14,XXREAL_0:2; then
      2 * (-1) to_power n < ((tn-tbn)/sqrt 5) + Lucas n/5 by FIB_NUM:7; then
      2 * (-1) to_power n < ((tn-tbn)/sqrt 5) + (tn+tbn)/5 by FIB_NUM3:21; then
      (2 * (-1) to_power n) * 10 < (((tn-tbn) / sqrt 5) + (tn+tbn) / 5) * 10
      by XREAL_1:68; then
      20 * (-1) to_power n < ((tn-tbn) / sqrt 5) * (2*5) + (tn+tbn) * 2; then
      20 * (-1) to_power n < ((tn-tbn) / sqrt 5) * (2*(sqrt 5)^2) + (tn+tbn)*2
      by SQUARE_1:def 2; then
      20*(-1) to_power n<((tn-tbn)/sqrt 5)*sqrt 5*(sqrt 5*2)+(tn+tbn)*2; then
      20*(-1) to_power n<(tn-tbn)*(2*sqrt 5)+(tn+tbn)*2 by A2,XCMPLX_1:87; then
      20 * (-1) to_power n + (5*(tn)^2 + 5*(tbn)^2 + 1) <
      (tn-tbn)*(2*sqrt 5)+(tn+tbn)*2+(5*(tn)^2+5*(tbn)^2+1) by XREAL_1:6; then
      20 * (-1) to_power n + (5*(tn)^2 + 5*(tbn)^2 + 1) -(tn+tbn)*2 <
      (tn-tbn)*(2*sqrt 5) + (tn+tbn)*2 + (5*(tn)^2 + 5*(tbn)^2 + 1)-(tn+tbn)*2
      by XREAL_1:9; then
      10*(-1) to_power n + 10 * (-1) to_power n + 5*(tn)^2 + 5*(tbn)^2 + 1 -
      (tn+tbn)*2 -10*(-1) to_power n <
      (tn-tbn)*(2*sqrt 5) + (5*(tn)^2 + 5*(tbn)^2 + 1)-10*(-1) to_power n
      by XREAL_1:9; then
      5*(2 * (tau*tau_bar) to_power n + (tn)^2 + (tbn)^2) + 1 -2*tn-2*tbn <
      (2*sqrt 5)*tn-(2*sqrt 5)*tbn + 5*(tn)^2 + 5*(tbn)^2 -
      10*(-1) to_power n + 1 by Lm3; then
      5*(2 * (tn *tbn) + (tn)^2 + (tbn)^2) + 1 -2*tn-2*tbn <
      (2*sqrt 5)*tn-(2*sqrt 5)*tbn + 5*(tn)^2 + 5*(tbn)^2 -
      5*2*(-1) to_power n + 1 by NEWTON:7; then
      5*(tn+tbn)^2 + 1 -2*tn-2*tbn <(2*sqrt 5)*tn-(2*sqrt 5)*tbn +
      5*((tn)^2 +(tbn)^2 -2*(tau*tau_bar) to_power n) + 1
      by Lm3; then
      5 * (tn + tbn)^2 +1 - 2 * tn - 2 * tbn < (2*sqrt 5)*tn - (2*sqrt 5)*tbn +
      5 * ((tn)^2 + (tbn)^2 - 2 * (tn * tbn)) + 1 by NEWTON:7; then
A15:   5 * (tn + tbn)^2 + 1 - 2 * (tn + tbn) < (2 * sqrt 5) * tn -
      (2 * sqrt 5) * tbn + 5 * (tn - tbn)^2 + 1;
A16:   5 * (Lucas n) ^2 + 1 - 2 * Lucas n >= 0
      proof
        5 * Lucas n >= 5 * 4 by A12,XREAL_1:66; then
        5 * Lucas n - 2 >= 20 - 2 by XREAL_1:9; then
        Lucas n * (5 * Lucas n - 2) >= 4 * 18 by A12,XREAL_1:66; then
        Lucas n * (5 * Lucas n - 2) + 1 >= 4 * 18 + 1 by XREAL_1:6;
        hence thesis;
      end;
A17:   2 * tau to_power (n+1) - tn + 2 * tau_bar to_power (n+1) - tbn + 1 > 0
      proof
        2 * tau to_power (n+1) - tn + 2 * tau_bar to_power (n+1) - tbn + 1 =
        2*(tn*tau to_power 1)-tn+2*tau_bar to_power (n+1)-tbn + 1 by Th2
        .= 2*(tn*tau to_power 1)-tn+2*(tbn*tb to_power 1)-tbn + 1 by Th2
        .= 2*(tn*tau) - tn + 2*(tbn*tb to_power 1) - tbn + 1
        .= tn * sqrt 5 + 2 * (tbn * tb) - tbn + 1 by FIB_NUM:def 1
        .= sqrt 5 * (tn - tbn) * 1 + 1 by FIB_NUM:def 2
        .= sqrt 5 * (tn - tbn) * (sqrt 5 / sqrt 5) + 1 by A2,XCMPLX_1:60
        .= sqrt 5 * (tn - tbn) * (sqrt 5 * (1 / sqrt 5)) + 1 by XCMPLX_1:99
        .= sqrt 5 * ((tn - tbn) * (1 / sqrt 5)) * sqrt 5 + 1
        .= sqrt 5 * ((tn - tbn) / sqrt 5) * sqrt 5 + 1 by XCMPLX_1:99
        .= sqrt 5 * Fib n * sqrt 5 + 1 by FIB_NUM:7
        .= (sqrt 5) ^2 * Fib n + 1
        .= 5 * Fib n + 1 by SQUARE_1:def 2;
        hence thesis;
      end;
      (2*sqrt 5) *tn - (2*sqrt 5) *tbn + 5 * (tn-tbn) ^2 + 1 = 2*sqrt 5*tn -
      2 * sqrt 5 * tbn + (sqrt 5) ^2 * (tn - tbn) ^2 + 1 by SQUARE_1:def 2
      .= (2 * tau * tn - 1 * tn + 2 * tb * tbn - 1 * tbn + 1) ^2
      by FIB_NUM:def 1,def 2
      .= (2*tau to_power 1 * tn - 1*tn + 2*tb*tbn - 1*tbn + 1) ^2
      .= (2 * (tau to_power 1*tn)-1*tn + 2*tb to_power 1*tbn - 1*tbn + 1) ^2
      .= (2*tau to_power (n+1)-tn+2*(tb to_power 1*tbn)-tbn+1)^2 by Th2
      .= (2*tau to_power (n+1) - tn + 2*tau_bar to_power (n+1) - tbn + 1) ^2
      by Th2; then
      5 * (Lucas n) ^2 + 1 - 2 * (tn+tbn) < (2 * tau to_power (n+1) - tn +
      2 * tau_bar to_power (n+1) - tbn + 1) ^2 by A15,FIB_NUM3:21; then
      5 * (Lucas n) ^2 + 1 - 2 * Lucas n < (2 * tau to_power (n+1) - tn +
      2 * tau_bar to_power (n+1) - tbn + 1) ^2 by FIB_NUM3:21; then
      sqrt (5*(Lucas n)^2 + 1 - 2*Lucas n) < sqrt ((2*tau to_power (n+1) - tn +
      2 * tau_bar to_power (n+1) - tbn + 1) ^2) by A16,SQUARE_1:27; then
      sqrt (5 * (Lucas n) ^2 + 1 - 2 * Lucas n) <
      2 * (tau to_power (n+1) + tau_bar to_power (n+1)) - tn
      - tbn + 1 by A17,SQUARE_1:22; then
      sqrt (5 *(Lucas n) ^2 - 2 * Lucas n + 1) < 2 *Lucas (n+1) - tn - tbn + 1
      by FIB_NUM3:21; then
      1 + sqrt (5*(Lucas n)^2 - 2*Lucas n + 1) < 2*Lucas (n+1) + 1 -tn - tbn +1
      by XREAL_1:6; then
      tn + tbn + 1 + sqrt (5 * (Lucas n) ^2 - 2 * Lucas n + 1) - tn - tbn <
      2 * Lucas (n+1) + 2 - tn - tbn; then
      Lucas n + 1 + sqrt (5 * (Lucas n) ^2 - 2 * Lucas n + 1) - tn - tbn <
      2 * Lucas (n+1) + 2 - tn - tbn by FIB_NUM3:21; then
      Lucas n + 1 + sqrt (5 * (Lucas n) ^2 - 2 * Lucas n + 1) - tn <
      2 * Lucas (n+1) + 2 - tn by XREAL_1:9; then
      Lucas n + 1 + sqrt (5 * (Lucas n) ^2 - 2 * Lucas n + 1) - 2 + 2 <
      2 * Lucas (n+1) + 2 by XREAL_1:9; then
      Lucas n + 1 + sqrt (5 * (Lucas n) ^2 - 2 * Lucas n + 1) - 2 <
      2 * Lucas (n+1) by XREAL_1:6; then
      ((Lucas n + 1 + sqrt (5 * (Lucas n) ^2 - 2 * Lucas n + 1)) - 2) / 2 <
      (2 * Lucas (n+1)) / 2 by XREAL_1:74;
      hence thesis;
    end;
    hence thesis by A3,INT_1:def 6;
  end;
