let n, m be Nat; :: thesis: ( m is odd & n >= m implies tau_bar to_power n >= tau_bar to_power m )
assume A1: m is odd ; :: thesis: ( not n >= m or tau_bar to_power n >= tau_bar to_power m )
assume n >= m ; :: thesis: tau_bar to_power n >= tau_bar to_power m
then A2: n + 1 > m + 0 by XREAL_1:8;

per cases ( n = m or n > m ) by A2, NAT_1:22;

suppose n = m ; :: thesis: tau_bar to_power n >= tau_bar to_power m

hence tau_bar to_power n >= tau_bar to_power m ; :: thesis: verum

end;

suppose A3: n > m ; :: thesis: tau_bar to_power n >= tau_bar to_power 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 tau_bar to_power n >= tau_bar to_power m ; :: thesis: verum

end;

end;