
theorem Th11:
  for n,m being Nat st n is even & m is even & n >= m holds
  tau_bar to_power n <= tau_bar to_power m
  proof
    let n,m be Nat;
    assume A1: n is even & m is even & 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 Th6,A1;
      n - m > m - m by A3,XREAL_1:9; then
      tau_bar to_power t < 1 by Th8,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;
