
theorem Th10:
  for n,m being Nat st m is odd & n >= m holds
  tau_bar to_power n >= tau_bar to_power m
  proof
    let n,m be Nat;
    assume A1: m is odd;
    assume n >= m; then
A2: n + 1 > m + 0 by XREAL_1:8;
    per cases by A2,NAT_1:22;
    suppose n = m;
      hence thesis;
    end;
    suppose A3: n > m; then
    reconsider t = n-m as Nat by NAT_1:21;
A4: tau_bar to_power n - tau_bar to_power m =
    tau_bar to_power (t + m) - tau_bar to_power m
    .= tau_bar to_power t * tau_bar to_power m - 1 * tau_bar to_power m
    by Th2
    .= (tau_bar to_power t - 1) * tau_bar to_power m;
A5: tau_bar to_power m <= 0 by Th7,A1;
 t <> 0 by A3;
    then |.tau_bar.| > 0 & t > 0; then
    (|.tau_bar.|) to_power t < 1 to_power t by Th5,POWER:37; then
    (|.tau_bar.|) to_power t < 1; then
    |.tau_bar to_power t.| < 1 & tau_bar to_power t <=
    |.tau_bar to_power t.| by ABSVALUE:4,SERIES_1:2; then
    tau_bar to_power t < 1 by XXREAL_0:2; then
    tau_bar to_power t - 1 <= 1 - 1 by XREAL_1:9; then
    tau_bar to_power n - tau_bar to_power m + tau_bar to_power m >=
    0 + tau_bar to_power m by A4,A5,XREAL_1:6;
    hence thesis;
    end;
  end;
