
theorem Th8:
  for n being Nat st n <> 0 holds tau_bar to_power n < 1/2
  proof
    let n be Nat;
    assume n <> 0; then
    n + 1 > 0 + 1 by XREAL_1:8; then
    n + 1 >= 1 + 1 by NAT_1:13; then
A1: n + 1 = 2 or n >= 2 by NAT_1:8;
    per cases by A1;
    suppose n = 1;
      hence thesis;
    end;
    suppose n >= 2; then
    n <> 0 & n <> 1; then
A2: n is non trivial Nat by NAT_2:def 1;
    (tau_bar) to_power n < 1/2
    proof
      defpred P[Nat] means (|.tau_bar.|) to_power $1 < 1/2;
A3:   (|.tau_bar.|) to_power 2 = (- tau_bar) to_power 2 by ABSVALUE:def 1
        .= (- tau_bar) ^2 by POWER:46
        .= (1 ^2 - 2 * 1 * sqrt 5 + (sqrt 5) ^2) / 4 by FIB_NUM:def 2
        .= (1 - 2 * sqrt 5 + 5) / 4 by SQUARE_1:def 2
        .= (3 - sqrt 5) / 2;
      sqrt 5 > 2 by SQUARE_1:20,27; then
      - sqrt 5 < - 2 by XREAL_1:24; then
      - sqrt 5 + 3 < - 2 + 3 by XREAL_1:8; then
A4:   P[2] by A3,XREAL_1:74;
A5:   for k being non trivial Nat st P[k] holds P[k+1]
      proof
        let k be non trivial Nat;
        assume P[k]; then
        (|.tau_bar.| to_power k) * (|.tau_bar.|) < (1/2) * (|.tau_bar.|)
        by XREAL_1:68; then
A6:     (|.tau_bar.| to_power k) * (|.tau_bar.| to_power 1) <
        (1/2) * (|.tau_bar.|);
        (1/2) * (|.tau_bar.|) < (1/2)*1 by Th5,XREAL_1:68; then
        (|.tau_bar.| to_power k) * (|.tau_bar.| to_power 1) < 1/2
          by A6,XXREAL_0:2;
        hence thesis by Th2;
      end;
A7:  for n being non trivial Nat holds P[n] from NAT_2:sch 2(A4, A5);
      for n being non trivial Nat holds tau_bar to_power n < 1/2
      proof
        let n be non trivial Nat;
        (|.tau_bar.|) to_power n < 1/2 by A7; then
        |.tau_bar to_power n.| < 1/2 &
        tau_bar to_power n <= |.tau_bar to_power n.|
        by ABSVALUE:4,SERIES_1:2;
        hence thesis by XXREAL_0:2;
      end;
      hence thesis by A2;
    end;
    hence thesis;
    end;
  end;
