
theorem Th14:
  for n being Nat st n > 1 holds -1/2 < tau_bar to_power n
  proof
    let n be Nat;
    assume A1: n > 1;
A2: n + 1 > 1 + 1 by A1,XREAL_1:8;
    per cases;
    suppose n is even; then
      consider k being Nat such that
A3:   n = 2 * k by ABIAN:def 2;
A4:    0^2 = 0;
      tau_bar to_power n = tau_bar to_power k to_power 2 by A3,NEWTON:9
      .= (tau_bar to_power k) ^2 by POWER:46;
      hence thesis by A4,SQUARE_1:12;
    end;
    suppose A5: n is odd;
      n >= 2 by A2,NAT_1:13; then
      n = 2 or n > 2 by XXREAL_0:1; then
      n + 1 > 2 + 1 by A5,POLYFORM:5,XREAL_1:6; then
      n >= 3 by NAT_1:13; then
A6:   tau_bar to_power n >= tau_bar to_power 3 by Th10,POLYFORM:6;
      sqrt 5 < sqrt ((5 / 2) ^2) by SQUARE_1:27; then
      sqrt 5 < 5 / 2 by SQUARE_1:def 2; then
      2 * sqrt 5 < 2 * (5 / 2) by XREAL_1:68; then
      -2 * sqrt 5 > - 5 by XREAL_1:24; then
      - 2 * sqrt 5 + 4 > -5 + 4 by XREAL_1:6; then
      (2 * (2 - sqrt 5)) / 2 > (-1) / 2 by XREAL_1:74;
      hence thesis by A6,Lm8,XXREAL_0:2;
    end;
  end;
