# Clean way to write a two column proof

I want to write proofs in the following format. The left-column I use for the exercise and the solution. And in the right-column I write results that I used in my proof. For example:

I'm able to do this, but I'm wondering if I can write this a little bit cleaner. Ideally I would like to have something like:

\begin{exercise}
\begin{question}
The exercise. [in the left-column]
\end{question}
\begin{proof}
The proof. [also in the left-column]
\end{proof}
\begin{results}
Some used results. [in the right-column]
\end{results}
\end{exercise}


At this moment, the code of the image look like this:

\documentclass[11pt]{article}
\usepackage[utf8]{inputenc}
\usepackage{amsmath, amssymb, amsthm}
\usepackage{framed}
\theoremstyle{definition}
\newtheorem{ex}{Exercise}
\usepackage[top=0.4in, bottom=0.4in, left=0.4in, right=0.4in]{geometry}

%preserves parkskip in the minipage
\usepackage{parskip}
\setlength{\parskip}{\medskipamount}
\makeatletter
\newcommand{\@minipagerestore}{\setlength{\parskip}{\medskipamount}}
\makeatother

\begin{document}
\begin{minipage}[t]{0.55\textwidth}
\begin{ex}
Show that $$e^ze^w=e^{z+w}$$ for all $$z,w\in\mathbb{C}$$.
\end{ex}

\begin{proof}
Let $$z,w\in \mathbb{C}$$. We now have that
\begin{align*}
e^ze^w&=\sum _{n=0}^\infty \frac{z^n}{n!}\sum _{n=0}^\infty \frac{w^n}{n!}
\\&=\sum _{n=0}^\infty \bigg(\sum _{k=0}^n \frac{z^k}{k!}\cdot \frac{w^{n-k}}{(n-k)!}\bigg)
\\& = \sum _{n=0}^\infty  \frac{1}{n!} \bigg(\sum _{k=0}^n {n\choose k} z^k w^{n-k}\bigg)
\\&=\sum _{n=0}^\infty  \frac{(z+w)^n}{n!}
\\&= e^{z+w}.
\end{align*}
\end{proof}
\end{minipage}
\hfill
\begin{minipage}[t]{0.4\textwidth}
{\abovedisplayskip=0pt\abovedisplayshortskip=0pt\belowdisplayskip=0pt\belowdisplayshortskip=0pt

\begin{framed}
\begin{equation*}
\sum _{n=0}^\infty a_{n} \sum _{n=0}^\infty b_{n}=\sum _{n=0}^\infty \sum _{k=0}^na_{k}b_{n-k}
\end{equation*}
\end{framed}

\begin{framed}
\begin{equation*}
(x+y)^n=\sum _{k=0}^n {n\choose k}x^k y^{n-k}
\end{equation*}
\end{framed}

\begin{framed}
\begin{equation*}
{n\choose k}=\frac{n!}{k!\cdot (n-k)!}
\end{equation*}
\end{framed}

\begin{framed}
\begin{equation*}
e^z=\sum _{n=0}^\infty  \frac{z^n}{n!} \qquad \forall z \in \mathbb{C}
\end{equation*}
\end{framed}
}
\end{minipage}


You can try using paracol package. It does pretty good job in making parallel multicolumn texts