
theorem
  for n being Nat st n <> 0 holds
    [\ (tau to_power (2*n)) / sqrt 5 /] = Fib (2*n)
    proof
      let n be Nat;
      assume A1: n <> 0;
A2:   (tau to_power (2*n)) / s5 - 1 < Fib (2*n)
      proof
A3:     2 to_power (2*n) > 0 by POWER:34;
        (1 - s5) to_power 2 to_power n < (2 to_power 2) to_power n
        by A1,Lm17,Lm19,POWER:37; then
        ((1-s5) to_power 2) to_power n < 2 to_power (2*n) by POWER:33; then
        (1 - s5) to_power (2*n) < 2 to_power (2*n) by NEWTON:9; then
        ((1 - s5) to_power (2*n)) / (2 to_power (2*n)) <
        (2 to_power (2*n)) / (2 to_power (2*n)) by A3,XREAL_1:74; then
        ((1 - s5) to_power (2*n)) / (2 to_power (2*n)) < 1
        by A3,XCMPLX_1:60; then
        tau_bar to_power (2*n) < 1 by Th1,FIB_NUM:def 2; then
        tau_bar to_power (2*n) < s5 by Lm16,XXREAL_0:2; then
        (tau_bar to_power (2*n))/s5 < s5 /s5 by Lm18,XREAL_1:74; then
        (tau_bar to_power (2*n)) / s5 < 1 by Lm18,XCMPLX_1:60; then
        - ((tau_bar to_power (2*n)) / s5) > -1 by XREAL_1:24; then
        - ((tau_bar to_power (2*n))/s5) + (tau to_power (2*n)) / s5 >
        -1 + (tau to_power (2*n)) / s5 by XREAL_1:8; then
        (tau to_power (2*n)) / s5 - ((tau_bar to_power (2*n))/s5) >
        -1 + (tau to_power (2*n)) / s5; then
        (tau to_power (2*n) - tau_bar to_power (2*n)) / s5 >
        -1 + (tau to_power (2*n)) / sqrt 5 by XCMPLX_1:120;
        hence thesis by FIB_NUM:7;
      end;
      tau_bar to_power (2*n) = tau_bar to_power 2 |^ n by NEWTON:9
      .= tau_bar ^2 to_power n by POWER:46; then
      tau_bar to_power (2*n) > 0 by POWER:34; then
      - (tau_bar to_power (2*n)) / sqrt 5 + (tau to_power (2*n)) / s5 <
      0 + (tau to_power (2*n)) / s5 by Lm18,XREAL_1:8; then
      (tau to_power (2*n)) / s5 - (tau_bar to_power (2*n)) / s5 <
      (tau to_power (2*n)) / s5; then
      (tau to_power (2*n) - tau_bar to_power (2*n)) / s5 <
      (tau to_power (2*n)) / s5 by XCMPLX_1:120; then
      Fib (2*n) <= (tau to_power (2*n)) / s5 by FIB_NUM:7;
      hence thesis by A2,INT_1:def 6;
   end;
