Real digital filter property I am a beginner to study about the filter notion and property

Being a real digital filter, 
(here "real filter" I means that its impulse response is real-valued)

this formula is established. 
But I have no idea how to prove it

$$ |H(\pi +w )|=|H(\pi -w)|$$

What should be $H(w)$ or $|H(w)| $ for generalized proof procedure?
HINT:

From the definition of the DTFT

$$H(\omega)=\sum_{n=-\infty}^{\infty}h[n]e^{-jn\omega}\tag{1}$$

derive the following facts:


$H(\omega)=H(\omega+2\pi)$
$H(\omega)=H^*(-\omega)$ for real-valued $h[n]$


where $*$ means complex conjugation. Combine these two results to prove the given equation.