2

How can you add vertical space between the word "Proof" and the actual start of the proof? This question shows how to start the proof on a new line, but I can't add space properly.

1

You can use the optional argument for \\ (I used 3ex in my example, but change to the desired value):

\documentclass{article}
\usepackage{amsthm}
\usepackage{lipsum}% just to generate text for the example

\makeatletter
\renewenvironment{proof}[1][\proofname]{\par
  \pushQED{\qed}%
  \normalfont \topsep6\p@\@plus6\p@\relax
  \trivlist
  \item[\hskip\labelsep
        \itshape
    #1\@addpunct{.}]\mbox{}\\*[3ex]
}{%
  \popQED\endtrivlist\@endpefalse
}
\makeatother

\begin{document}

\begin{proof}
\lipsum*[1]
\end{proof}

\end{document}

enter image description here

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