Mathematical relationship between highpass and lowpass filtering
Yes you are right. The following line (excerpted from your post) is wrong, below which I place the corrected version:
\begin{align*}
&amp; g * h_{HP}\\
&amp; = \mathcal{F}^{-1}(G \cdot H_{HP})\\
&amp; = \mathcal{F}^{-1}(G \cdot (1 - H_{LP}))\\
&amp; = g * \mathcal{F}^{-1}(1 - H_{LP}) &amp;\quad \text{(The next step is unclear to me)}\\
&amp; = g * (1 - h_{LP}) &amp;\quad \text{(This line is wrong)}\\
&amp; = g * (\delta(t) - h_{LP}) &amp;\quad \text{(This is the corrected version)}\\
&amp; = g - (g * h_{LP})
\end{align*}

I don't think so if you really need to go deeper than the above corrected proof to show that the inital claim was right. But it's up to you of course.