Indent the word "Proof" in environment proof of amsthm

Like this?

\documentclass[11pt]{article}

\usepackage{amsthm,xpatch}

\xpatchcmd{\proof}{\hskip\labelsep}{\hskip5\labelsep}{}{}  %% change 5 here as you wish

\begin{document}
\begin{proof}
This is the proof of the lemma This is the proof of the lemma This is the proof of the lemma This is the proof of the lemma This is the proof of the lemma
\end{proof}

\end{document}

You have to load xpatch package (by adding \usepackage{xpatch} in the preamble) and then add the following to your preamble:

\xpatchcmd{\proof}{\hskip\labelsep}{\hskip5\labelsep}{}{}

If you don't have xpatch installed, then add the following to your preamble (after \usepackage{amsthm})

\makeatletter
\renewenvironment{proof}[1][\proofname]{\par
  \normalfont
  \topsep6\p@\@plus6\p@ \trivlist
  \item[\hskip5\labelsep\itshape
    #1\@addpunct{.}]\ignorespaces
}{%
  \qed\endtrivlist
}
\makeatother

Another way to accomplish this request is to adapt the definition of proof from amsproc.cls where it is already indented.

That definition inserts the line \itemindent\normalparindent between \trivlist and \item, and leaves \hskip\labelsep unchanged.

This also requires in the preamble, before redefining proof, insertion of a line \normalparindent\parindent.

(AMS document classes avoid this problem by establishing \normalparindent in the class file.)