
theorem Th15:
  for n being Nat st n > 2 holds tau_bar to_power n >= - 1/sqrt 5
  proof
    let n be Nat;
    assume  n > 2;
    then n >= 2+1 by NAT_1:13; then
A1: tau_bar to_power n >= tau_bar to_power 3 by Th10,POLYFORM:6;
A2: sqrt 5 > 0 by SQUARE_1:25;
    sqrt 5 >= 2 by SQUARE_1:20,26; then
    2 * sqrt 5 >= 2 * 2 by XREAL_1:64; then
    2 * sqrt 5 - 5 >= 4 - 5 by XREAL_1:9; then
    2 * sqrt 5 - (sqrt 5)^2 >= -1 by SQUARE_1:def 2; then
    ((2 - sqrt 5) * sqrt 5) / sqrt 5 >= (-1) / sqrt 5 by A2,XREAL_1:72; then
    ((2 - sqrt 5) * sqrt 5) / sqrt 5 >= -1 / sqrt 5 by XCMPLX_1:187; then
    2 - sqrt 5 >= -1 / sqrt 5 by A2,XCMPLX_1:89;
    hence thesis by A1,Lm8,XXREAL_0:2;
  end;
