
theorem
  for n being Nat st n <> 0 holds
    [/ (tau to_power n)/sqrt 5 - 1/2 \] = Fib n
  proof
    let n be Nat;
    set tn = tau to_power n;
    set tbn = tau_bar to_power n;
    assume A1: n <> 0;
A2: sqrt 5 > 0 by SQUARE_1:25;
A3: tn / sqrt 5 - 1/2 <= Fib n
    proof
      1 <= sqrt 5 by SQUARE_1:18,26; then
   1 / 2 <= sqrt 5 / 2 by XREAL_1:72;
      then tbn <= sqrt 5 / 2 by A1,Th8,XXREAL_0:2; then
      tbn / sqrt 5 <= sqrt 5 / 2 / sqrt 5 by A2,XREAL_1:72; then
      tbn / sqrt 5 <= sqrt 5 / sqrt 5 / 2 by XCMPLX_1:48; then
      tbn / sqrt 5 <= 1 / 2 by A2,XCMPLX_1:60; then
      -tbn / sqrt 5 >= -1 / 2 by XREAL_1:24; then
      -tbn / sqrt 5 + tn/sqrt 5 >= -1/2 + tn / sqrt 5 by XREAL_1:6; then
      tn / sqrt 5 - tbn / sqrt 5 >= -1/2 + tn / sqrt 5; then
      (tn - tbn) / sqrt 5 >= tn / sqrt 5 - 1/2 by XCMPLX_1:120;
      hence thesis by FIB_NUM:7;
    end;
    tn / sqrt 5 - 1 / 2 + 1 > Fib n
    proof
      n+1 > 0+1 by A1,XREAL_1:6; then
A4:   n >= 1 by NAT_1:13;
      per cases by A4,XXREAL_0:1;
      suppose A5: n = 1; then
A6:     tn / sqrt 5 - 1/2 + 1 = tau / sqrt 5 - 1/2 + 1
        .= (1 + sqrt 5) / 2 / sqrt 5 + (1 - 1/2) by FIB_NUM:def 1
        .= (1 + sqrt 5) / sqrt 5 / 2 + 1 / 2 by XCMPLX_1:48
        .= ((1 + sqrt 5) / sqrt 5 + 1) / 2
        .= (1 / sqrt 5 + sqrt 5 / sqrt 5 + 1) / 2 by XCMPLX_1:62
        .= (1 / sqrt 5 + 1 + 1) / 2 by A2,XCMPLX_1:60
        .= 1 / sqrt 5 / 2 + 2 / 2;
        1/sqrt 5/2 + 1 > 0 + 1 by A2,XREAL_1:6;
        hence thesis by A5,A6,PRE_FF:1;
      end;
      suppose A7: n > 1;
      1 < sqrt 5 by SQUARE_1:18,27; then
      1 / 2 < sqrt 5 / 2 by XREAL_1:74; then
A8:   - 1 / 2 > - sqrt 5 / 2 by XREAL_1:24;
      tbn > - 1 / 2 by A7,Th14; then
      tbn > - sqrt 5 / 2 by A8,XXREAL_0:2; then
      tbn / sqrt 5 > (-sqrt 5 / 2) / sqrt 5 by A2,XREAL_1:74; then
      tbn / sqrt 5 > - (sqrt 5 / 2 / sqrt 5) by XCMPLX_1:187; then
      tbn / sqrt 5 > - (sqrt 5 / sqrt 5 / 2) by XCMPLX_1:48; then
      tbn / sqrt 5 > - 1 / 2 by A2,XCMPLX_1:60; then
      -tbn / sqrt 5 < - -1 / 2 by XREAL_1:24; then
      - tbn / sqrt 5 + tn / sqrt 5 < 1 / 2 + tn / sqrt 5 by XREAL_1:6; then
      tn / sqrt 5 - tbn / sqrt 5 < 1 / 2 + tn / sqrt 5; then
      (tn - tbn) / sqrt 5 < 1 / 2 + tn / sqrt 5 by XCMPLX_1:120;
      hence thesis by FIB_NUM:7;
      end;
    end;
    hence thesis by A3,INT_1:def 7;
  end;
