Starting a proof with \paragraph
Add an \item or ~ before \paragraph. For example:
\begin{document}
\begin{proof}
\item
\paragraph{This bit in bold.} Now this part of the proof.
\end{proof}
\end{document}
I think the best way is to define a new environment, let's say breakproof
, that acts like proof
but starts with a new line after the word "Proof".
So you can write
\documentclass{article}
\usepackage{amsthm}
\makeatletter
\newenvironment{breakproof}[1][\proofname]{\par
\pushQED{\qed}%
\normalfont \topsep6\p@\@plus6\p@\relax
\trivlist
\item[\hskip\labelsep
\itshape
#1\@addpunct{.}]\ignorespaces\item
}{%
\popQED\endtrivlist\@endpefalse
}
\makeatother
\begin{document}
\begin{breakproof}
\textbf{This bit in bold.}
\newline
Now this part of the proof.
\end{breakproof}
\end{document}
and the result is:
Otherwise, you can redefine the proof
environment (bad practice, in my opinion) and use it as before (the result is the same):
\documentclass{article}
\usepackage{amsthm}
\makeatletter
\renewenvironment{proof}[1][\proofname]{\par
\pushQED{\qed}%
\normalfont \topsep6\p@\@plus6\p@\relax
\trivlist
\item[\hskip\labelsep
\itshape
#1\@addpunct{.}]\ignorespaces\item
}{%
\popQED\endtrivlist\@endpefalse
}
\makeatother
\begin{document}
\begin{proof}
\textbf{This bit in bold.}
\newline
Now this part of the proof.
\end{proof}
\end{document}