
theorem Th16:
  for n being Nat st n >= 2 holds tau_bar to_power n <= 1/sqrt 5
  proof
    let n be Nat;
    assume A1: n >= 2;
    per cases;
    suppose A2: n is even;
A3:   sqrt 5 > 0 by SQUARE_1:25;
A4:   tau_bar to_power n <= (3 - sqrt 5) / 2 by Lm7,A1,A2,Th11,POLYFORM:5;
      sqrt 5 <= sqrt ((7/3) ^2) by SQUARE_1:26; then
      sqrt 5 <= 7/3 by SQUARE_1:def 2; then
      3 * sqrt 5 <= (7 / 3) * 3 by XREAL_1:64; then
      3 * sqrt 5 - 5 <= 7 - 5 by XREAL_1:9; then
      3 * sqrt 5 - (sqrt 5) ^2 <= 2 by SQUARE_1:def 2; then
      (3 * sqrt 5 - sqrt 5 * sqrt 5) / 2 <= 2 / 2 by XREAL_1:72; then
      (((3-sqrt 5) / 2) * sqrt 5) / sqrt 5 <= 1 / sqrt 5 by A3,XREAL_1:72; then
      (3 - sqrt 5) / 2 <= 1 / sqrt 5 by A3,XCMPLX_1:89;
      hence thesis by A4,XXREAL_0:2;
    end;
    suppose n is odd; then
A5:   tau_bar to_power n < 0 by Th7;
      sqrt 5 >= sqrt 0 by SQUARE_1:26;
      hence thesis by A5;
    end;
  end;
