let n, m be natural number ; :: thesis: ( n is even & m is even & n >= m implies tau_bar to_power n <= tau_bar to_power m )
assume A0: ( n is even & m is even & n >= m ) ; :: thesis: tau_bar to_power n <= tau_bar to_power m
then ze: n + 1 > m + 0 by XREAL_1:10;

per cases ( n = m or n > m ) by ze, 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 A2: n > m ; :: thesis: tau_bar to_power n <= tau_bar to_power m

then reconsider t = n - m as natural number by NAT_1:21;
C2: (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 JakPower32
.= ((tau_bar to_power t) - 1) * (tau_bar to_power m) ;
C3: tau_bar to_power m > 0 by pom1, A0;
n - m > m - m by A2, XREAL_1:11;
then tau_bar to_power t < 1 by hop8, XXREAL_0:2;
then (tau_bar to_power t) - 1 < 1 - 1 by XREAL_1:11;
then ((tau_bar to_power n) - (tau_bar to_power m)) + (tau_bar to_power m) < 0 + (tau_bar to_power m) by C2, C3, XREAL_1:8;
hence tau_bar to_power n <= tau_bar to_power m ; :: thesis: verum

end;

end;