This is a variation on cmhughes's solution. It uses description
and enumerate
with enumitem
to define two new lists and some new commands to keep things consistent.
Anyway, the idea was to provide the following new commands:
\mythm
\myproofsketch
\myassume
\myprove
for starting the various parts of the proof. And the following environments:
mytheorem
to contain the whole thing and to list the assumptions with enumeration, respectively.
\documentclass{article}
\usepackage{enumitem}
\newlength{\mylstindent}
\settowidth\mylstindent{\bfseries Theorem}
\newlist{mytheoremlst}{description}{1}
\setlist[mytheoremlst]{leftmargin=0pt,itemindent=\mylstindent,style=sameline,font={\normalfont\textsc},noitemsep}
\newlist{myassumptions}{enumerate}{1}
\setlist[myassumptions]{nosep,label=\arabic*.,align=left,itemindent=\mylstindent,before={\hspace*{-\mylstindent}\vspace*{-1.2em}}}
\newcommand*{\myproofsketch}{%
\vskip 5pt
\item[Proof Sketch:]}
\newcommand*{\myassume}{\item[Assume:]\begin{myassumptions}}
\newcommand*{\myprove}{\end{myassumptions}\item[Prove:]}
\newcommand*{\mythm}{\item[\normalfont\bfseries Theorem]}
\newenvironment{mytheorem}{%
\begin{mytheoremlst}%
}%
{\end{mytheoremlst}%
}
\begin{document}
\begin{mytheorem}
\mythm There does not exist $r$ in $\mathbf{Q}$ such that $r^2=2$.
\myproofsketch We assume $r^2=2$ for $r \in \mathbf{Q}$ and argue to a \emph{reductio}. Writing $r=\frac{a}{b}$ where the HCD of $a$ and $b$ is $1$, we deduce from $\frac{m}{n}^2=2$\dots
\myassume%
\item $r \in \mathbf{Q}$
\item $r^2 = 2$
\myprove False
\end{mytheorem}
\end{document}
Produces:

Can anybody suggest a better way of setting up myassumption
to get the first assumption aligned vertically with 'Assume:' without losing the horizontal alignment with the second assumption? This is somewhat of a hack and likely fragile...