Insertion loss equality - proof From this paper (slide 11), the insertion loss (or attenuation) is defined as 

$$
L(\omega^2)=\frac{\lvert V_i(j\omega)\rvert^2}{\lvert V_o(j\omega)\rvert^2}=\frac{1}{\lvert H(j\omega)\rvert^2}=10 \log\left(\frac{1}{H(j\omega)H(-j\omega)}\right)
$$

I'm interested specifically in, where does the last inequality follow?
As pointed out in PeterK.'s comment, the way it is written this does not make sense. What is meant here is that the insertion loss can be measured on a linear scale (i.e., as a factor) as

$$L(\omega^2)=\frac{1}{|H(j\omega)|^2}$$

or it can be given in decibels:

$$L_{dB}(\omega^2))=10\log_{10}(L(\omega^2))$$

The remaining question could be why $|H(j\omega)|^2$ can be replaced by $H(j\omega)H(-j\omega)$. The reason is the symmetry of the frequency response $H(j\omega)$ for real-valued filters:

$$H(j\omega)=H^*(-j\omega)$$

From this it follows that

$$|H(j\omega)|^2=H(j\omega)H^*(j\omega)=H(j\omega)H(-j\omega)$$