or

${\displaystyle \int \ldots \int (\Delta \eta e^{\Delta \eta }-e^{\Delta \eta }+1)\,dp_{1}\ldots dq_{n}>0.}$

(454)

Since the parenthesis in this expression represents a positive value, except when it vanishes for ${\displaystyle \Delta \eta =0}$, the integral will be positive unless ${\displaystyle \Delta \eta }$ vanishes everywhere within the limits, which would make the difference of the two distributions vanish. The theorem is therefore proved.