
theorem
  for n being non zero Nat holds
    tau to_power n > tau_bar to_power n
  proof
    let n be non zero Nat;
    set tb = tau_bar;
    per cases;
    suppose A1: n is even;
      consider k being Nat such that A2: n = 2*k by A1,ABIAN:def 2;
A3:   k > 0 by A2;
A4:   tau to_power n = (tau to_power 2) to_power k by A2,NEWTON:9
      .= (tau ^2) to_power k by POWER:46;
      tau_bar to_power n = (tau_bar to_power 2) to_power k by A2,NEWTON:9
      .= (tau_bar)^2 to_power k by POWER:46;
      hence thesis by A3,A4,Lm14,POWER:37;
    end;
    suppose  n is odd;
      then
consider k being Nat such that A5: n = 2*k+1 by ABIAN:9;
      set kk = tau to_power (2*k);
A6:   (tau/tb) to_power (2*k) = (tau/tb) to_power 2 to_power k by NEWTON:9
      .= ((tau/tb) ^2) to_power k by POWER:46;
      tb to_power (2 * k) = (tb to_power 2) to_power k by NEWTON:9
      .= (tau_bar ^2) to_power k by POWER:46; then
A7:   tb to_power (2 * k) > 0 by POWER:34;
      (tau / tb) to_power (2*k) > (sqrt 5 - 3)/2 by A6,Lm15,POWER:34; then
      (tau / tb) to_power (2*k) * ((-3-sqrt 5)/2) < (tb/tau) * ((-3-sqrt 5)/2)
      by Lm13,Lm4,XREAL_1:69; then
      (tau/tb) to_power (2*k) * (tau/tb) < 1 by Lm5,XCMPLX_1:112; then
      (kk/tb to_power (2*k)) * (tau/tb) < 1 by Th1; then
      kk * (1/tb to_power (2*k)) * (tau/tb) < 1 by XCMPLX_1:99; then
      kk*(1/tb to_power (2*k)) * (tau/tb)*tb to_power (2*k) <
      1 * (tb to_power (2*k)) by A7,XREAL_1:68; then
      kk * (tau/tb)*(tb to_power (2*k) *(1/tb to_power (2*k)))
      < 1 * (tb to_power (2*k)); then
      kk * (tau/tb)*(tb to_power (2*k)/tb to_power (2*k)) <
      1 * (tb to_power (2*k)) by XCMPLX_1:99; then
      kk*(tau/tb)*1 < tb to_power (2*k) by A7,XCMPLX_1:60; then
      kk * (tau *(1/tb))*1 < tb to_power (2*k) by XCMPLX_1:99; then
      kk * (tau *(1/tb)) * 1 * tb > tb to_power (2*k) * tb by XREAL_1:69; then
      kk * tau * ((1/tb) *1 *tb) > tb to_power (2*k) * tb; then
      kk * tau * (tb / tb) > tb to_power (2*k) * tb by XCMPLX_1:99; then
      kk *tau*1 > tb to_power (2*k) * tb by XCMPLX_1:60; then
      kk*tau > tb to_power (2*k)*tb to_power 1;then
      kk * tau > tb to_power (2*k+1) by Th2; then
      kk * tau to_power 1 > tb to_power (2*k+1);
      hence thesis by A5,Th2;
    end;
  end;
