documentclass{article}
usepackage{lipsum}
usepackage{amsthm}
begin{document}
    begin{proof}
        lipsum [1]
    end{proof}
end{document}

documentclass{article}
usepackage{lipsum}
usepackage{amsthm}
newlengthmyproof
settowidthmyproof{Proof. }

begin{document}
    begin{proof}
        begin{minipage}[t]{dimexprlinewidth-myproof}
        lipsum [1]
        end{minipage}vskip-baselineskip
    end{proof}
end{document}