
theorem
  for n being Nat st n >= 2 holds
    Fib (n+1) = [\ (Fib(n) + 1 + sqrt(5*(Fib(n))^2 - 2 * Fib(n) + 1) )/2 /]
  proof
    let n be Nat;
    assume A1: n >= 2;
A2: n - 0 is Element of NAT by NAT_1:21;
A3: n - 1 >= 2 - 1 by A1,XREAL_1:9;
 (n + 1 -' 1) + 1 = (n + 1 - 1) + 1 by NAT_D:37
    .= (n - 1 + 1) + 1
    .= (n -' 1 + 1) + 1 by A3,NAT_D:39;
then A4: Fib (n+1) = Fib (((n-'1)+1)+1) by NAT_D:34
    .= Fib (n-'1) + Fib (n-'1+1) by PRE_FF:1
    .= Fib (n-'1) + Fib (n+1-'1) by A1,NAT_D:38,XXREAL_0:2
    .= Fib (n -' 1) + Fib (n) by NAT_D:34;
    n + 2 >= 2 + 2 by A1,XREAL_1:6; then
    reconsider p = n + 2 - 1 as Nat by NAT_1:21,XXREAL_0:2;
A5: Lucas (n) - Fib(n) = Lucas (n+1-'1) - Fib (n) by NAT_D:34
    .= Lucas (n-'1+1) - Fib n by A1,NAT_D:38,XXREAL_0:2
    .= Fib(n-'1) + Fib((n-'1)+2) - Fib(n) by FIB_NUM3:20
    .= Fib (n-'1) + Fib (n+2-'1) - Fib (n) by A1,NAT_D:38,XXREAL_0:2
    .= Fib (n-'1) + Fib p - Fib (n) by NAT_D:37
    .= 2 * Fib (n-'1) by A4;
A6: (Lucas (n))^2 - 5 * (Fib n) ^2 =
    (Lucas (n))^2 - 5 * (Fib n) to_power 2 by POWER:46
    .= (Lucas n) to_power 2 - 5 * (Fib n) |^2 by POWER:46
    .= (-1) * (5 * (Fib (n)) |^2 - (Lucas (n)) |^2)
    .= (-1) * (4 * (-1) to_power (n+1)) by A2,FIB_NUM3:30
    .= 4 * ((-1) * (-1) to_power (n+1))
    .= 4 * ((-1) to_power 1 * (-1) to_power (n+1))
    .= 4 * (-1) to_power (1+(n+1)) by Th2
    .= 4 * (-1) to_power (n+2)
    .= 4 * ((-1) to_power n * (-1) to_power 2) by Th2
    .= 4 * ((-1) to_power n * (-1) ^2) by POWER:46
    .= 4 * (-1) to_power n;
    n -' 1 >= 2 -'1 & 2-1>= 1 by A1,NAT_D:42; then
    n -'1 >= 2-1 by NAT_D:39; then
A7: Fib (n-'1) >= 1 by FIB_NUM2:45,PRE_FF:1;
    1 >= (-1) to_power n
    proof
      per cases;
      suppose n is even;
        hence thesis by FIB_NUM2:3;
      end;
      suppose n is odd;
        hence thesis by FIB_NUM2:2;
      end;
    end; then
    (-1) to_power n <= Fib (n-'1) by A7,XXREAL_0:2; then
    (Lucas n)^2 - 5 * (Fib n)^2 <= 2 * 2 * Fib (n-'1) by A6,XREAL_1:64; then
    (Lucas n)^2 - 5 * (Fib n)^2 - 2 * Lucas (n) <=
    2 * Lucas (n) - 2 * Fib (n) - 2 * Lucas (n)
      by A5,XREAL_1:9; then
    (Lucas n)^2 - 5 * (Fib n)^2 - 2 * Lucas (n) + 1 <= - 2 * Fib (n) + 1
      by XREAL_1:6; then
A8: (Lucas n)^2 - 5 * (Fib n)^2 - 2 * Lucas (n) + 1 + 5 * (Fib n)^2 <=
    - 2 * Fib (n) + 1 + 5 * (Fib n)^2 by XREAL_1:6;
A9: n + 2 -' 1 = n + 2 - 1 by NAT_D:37 .= n + 1;
A10: Lucas n = Lucas (n+1-'1) by NAT_D:34
    .= Lucas (n -' 1 + 1) by A1,NAT_D:38,XXREAL_0:2
    .= Fib (n -' 1) + Fib ((n -' 1) + 2) by FIB_NUM3:20
    .= Fib (n -' 1) + Fib (n + 2 -' 1) by A1,NAT_D:38,XXREAL_0:2
    .= 2 * Fib (n + 1) - Fib n by A4,A9;
A11: 2 * Fib (n-'1) >= 2 * 0;
    n >= 1 by A1,XXREAL_0:2; then
A12: Fib n >= 1 by FIB_NUM2:45,PRE_FF:1; then
    -Fib n <= -1 by XREAL_1:24; then
    -Fib n + 1 <= -1 + 1 by XREAL_1:6; then
    2 * Fib (n -'1) + Fib n >= - Fib n + 1 + Fib n by A11,XREAL_1:6; then
A13: 2 * Fib (n-'1) + Fib n - 1 >= 1 - 1 by XREAL_1:9;
    sqrt ((2 * Fib (n+1) - Fib n - 1) ^2) <= sqrt (5*(Fib n)^2 - 2 * Fib n + 1)
    by A10,A8,SQUARE_1:26; then
    2 * Fib (n+1) - Fib n - 1 <= sqrt (5*(Fib n) ^2 - 2 * Fib n + 1)
    by A4,A13,SQUARE_1:22; then
    2 * Fib (n+1) - Fib n - 1 + Fib n <=
    sqrt (5 * (Fib n) ^2 - 2 * Fib n + 1) + Fib n by XREAL_1:6; then
    2 * Fib (n+1) - Fib n - 1 + Fib n + 1 <=
    sqrt (5*(Fib n) ^2 - 2 * Fib n + 1) + Fib n + 1 by XREAL_1:6; then
A14: (2 * Fib (n+1)) / 2 <=
    (sqrt (5*(Fib n) ^2 - 2 * Fib n + 1) + Fib n + 1) / 2 by XREAL_1:72;
A15: 5 * (Fib n) ^2 - 2 * Fib n = (5 * Fib n - 2) * Fib n;
    5 * Fib n >= 5 * 1 by A12,XREAL_1:64; then
    5 * Fib n - 2 >= 5 - 2 by XREAL_1:9; then
A16: 5 * Fib n - 2 >= 0 by XXREAL_0:2;
A17: Fib (n+1) + Fib (n+1) - Fib n + 1 = Fib (n+1) + Fib (n-'1) + 1 by A4;
    (Lucas n) ^2 - 5*(Fib n) ^2 > -2 * (Lucas n + Fib n)
    proof
A18:   Fib n > 0 by A1,FIB_NUM2:21,45;
      Lucas n >= n by FIB_NUM3:17; then
      Lucas n >= 2 by A1,XXREAL_0:2; then
      Lucas n + Fib n > 0 + 2 by A18,XREAL_1:8; then
      (Lucas n + Fib n) * 2 > 2 * 2 by XREAL_1:68; then
A19:   -(Lucas n + Fib n) * 2 < -4 by XREAL_1:24;
      -4 <= 4 * (-1) to_power n by Lm1;
      hence thesis by A6,A19,XXREAL_0:2;
    end; then
    (Lucas n) ^2 - 5*(Fib n) ^2 + 2 * Lucas (n) >
    -2 * (Lucas n + Fib n) + 2 * Lucas (n) by XREAL_1:6; then
    (Lucas n) ^2 - 5 * (Fib n) ^2 + 2 * Lucas n + 5 * (Fib n) ^2 >
    -2 * Fib n + 5 * (Fib n) ^2 by XREAL_1:6; then
    (Lucas n) ^2 + 2* Lucas (n) * 1 + 1 ^2 > -2 * Fib n + 5 * (Fib n) ^2 + 1 ^2
    by XREAL_1:6; then
    sqrt((2 * Fib (n+1)- Fib n + 1) ^2) > sqrt(-2 * Fib n + 5*(Fib n) ^2 + 1)
    by A16,A15,A10,SQUARE_1:27; then
    2*Fib (n+1)-Fib n+1  > sqrt(-2*Fib n+5*(Fib n)^2+1) by A17,SQUARE_1:22;
 then
    2*Fib (n+1)-Fib n+1-1  > sqrt(-2*Fib n+5*(Fib n)^2+1)-1 by XREAL_1:9; then
    2 * Fib (n+1) - Fib n + Fib n  >
    sqrt(-2 * Fib n + 5 * (Fib n) ^2 + 1) - 1 + Fib n by XREAL_1:6; then
    (2 * Fib (n+1) ) / 2  >
    (sqrt(-2 * Fib n + 5*(Fib n)^2 + 1) - 1 + Fib n) / 2 by XREAL_1:74; then
    (Fib n + 1 + sqrt (5*(Fib(n)) ^2 - 2 * Fib n + 1)) / 2 - 1 < Fib (n+1);
    hence thesis by A14,INT_1:def 6;
  end;
