\documentclass{report}

\usepackage{amsmath,amsthm}

\setlength{\parindent}{0pt}

\newtheoremstyle{mytheoremstyle}{0pt}{0pt}{\itshape}{}{\bfseries}{.}{.5em}{}

\theoremstyle{mytheoremstyle}

\newtheorem{theorem}{Theorem}[chapter]
\newtheorem{lemma}[theorem]{Lemma}

\AtBeginDocument{%
  \setlength{\abovedisplayskip}{0pt}%
  \setlength{\belowdisplayskip}{0pt}%
  \setlength{\abovedisplayshortskip}{0pt}%
  \setlength{\belowdisplayshortskip}{0pt}%
}

\begin{document}
Some normal text goes here, with a lemma immediately below.

\begin{lemma}
This is the statement of the lemma.
\begin{align*}
  x_1 + x_2 + x_3 + x_4                        &\leq 10 \\
                         x_5 + x_6             &\leq 8 \\
                                    x_7 + x_8  &\leq 5
\end{align*}
\end{lemma}
\begin{proof}
It is possible to derive the following:
This is the proof of the lemma, with a mysterious space above.
\end{proof}

Some normal text goes here, with a mysterious space above.

The next paragraph comes immediately below.

\end{document}