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\begin{thm}
Quisque aliquam $x$ ipsum sed turpis.
Pellentesque $y\in K$ laoreet velit nec justo.
Nam sed augue.
Maecenas rutrum quam eu dolor.
\begin{equation} 
  \int_5^6 x^2\,dx=A_{xt}  \label{eq:first}
\end{equation}
Fusce consectetuer.
Proin tellus est, luctus vitae, molestie a, mattis et, mauris.
\begin{equation}\begin{split}
  H_c&=\frac{1}{2n} \sum^n_{l=0}(-1)^{l}(n-{l})^{p-2}
  \sum_{l _1+\dots+ l _p=l}\prod^p_{i=1} \binom{n_i}{l _i}\\
  &\quad\cdot[(n-l )-(n_i-l _i)]^{n_i-l _i}\cdot
  \Bigl[(n-l )^2-\sum^p_{j=1}(n_i-l _i)^2\Bigr].
\end{split}\end{equation}
Donec tempor. 
Pellentesque habitant morbi tristique senectus et netus et malesuada 
fames ac turpis egestas.
\end{thm}

\begin{proof}
Fusce adipiscing justo nec ante. 
Nullam in enim equation~\ref{eq:first}.
\begin{equation*}
  \left.\begin{aligned}
  B'&=-\partial\times E,\\
  E'&=\partial\times B - 4\pi j
  \end{aligned}
  \right\}
  \qquad \text{Maxwell's equations}
\end{equation*}
Pellentesque felis orci, sagittis ac, malesuada et, facilisis in, ligula. 
Nunc non magna sit amet mi aliquam dictum. 
\begin{equation}
  \frac{1}{k}\log_2 c(f)\quad\tfrac{1}{k}\log_2 c(f)\quad
  \sqrt{\frac{1}{k}\log_2 c(f)}\quad\sqrt{\dfrac{1}{k}\log_2 c(f)}
\end{equation}
In mi.
\end{proof}

\lipsum[26]

\begin{defn}
Aenean adipiscing auctor est. Morbi quam
arcu, malesuada sed, volutpat et, elementum sit amet, libero. Duis
accumsan. Curabitur urna.
\begin{equation}
  \begin{pmatrix} a&b&c&d\\
  e&\hdotsfor{3} \end{pmatrix}
\end{equation}
In sed ipsum.
\end{defn}

\begin{lem}
Donec lobortis nibh. 
Duis $x\in K_2$ mattis. 
Sed cursus lectus quis odio. 
Phasellus arcu. 
Praesent imperdiet dui in sapien. 
\end{lem}

\begin{proof}
Vestibulum tellus pede, auctor a, pellentesque sit amet, vulputate sed, purus. 
\begin{align}
  A_1&=N_0(\lambda;\Omega')-\phi(\lambda;\Omega'),\\
  A_2&=\phi(\lambda;\Omega')-\phi(\lambda;\Omega),\\
  \intertext{and}
  A_3&=\mathcal{N}(\lambda;\omega).
\end{align}
Nunc pulvinar, dui at eleifend adipiscing, tellus nulla placerat massa, 
sed condimentum nulla tellus sed ligula. 
Nulla vitae odio sit amet leo imperdiet blandit. 
In vel massa.
\begin{equation*}
  \sum_{\begin{subarray}{l}
        i\in\Lambda\\ 0<j<n
      \end{subarray}}
   P(i,j)
\end{equation*}
Maecenas varius dui at turpis. 
Sed odio.
\end{proof}

\begin{thm}
Sed justo. 
Maecenas lacinia, turpis sed commodo congue, odio urna elementum nunc, 
vitae molestie velit nunc eu sem. 
Maecenas enim.
\begin{equation}
  \displaystyle
  \sum_{\substack{0\le i\le m\\ 0<j<n}} P(i,j)
\end{equation}
Proin quis neque nec tortor sollicitudin volutpat.
\end{thm}

\begin{proof} 
Sed at ante. 
Sed vitae mauris non ante egestas hendrerit.
Cum sociis natoque penatibus et magnis dis parturient montes, nascetur 
ridiculus mus. 
\begin{equation}\label{xx}
  \begin{split}
  a& =b+c-d\\
   & \quad +e-f\\
   & =g+h\\
   & =i
  \end{split}
\end{equation}
\end{proof}

\begin{cor}
In venenatis $2\leq y,z\leq 5$ facilisis magna.
Cras quis mauris. 
Aliquam eget
\begin{math}
  \bigl( \begin{smallmatrix}
    a&b\\ c&d
  \end{smallmatrix} \bigr)
\end{math}
magna. 
Donec rutrum sagittis mi. 
Morbi elementum, est sit amet sollicitudin feugiat, orci magna semper risus, 
eu congue nulla metus vel elit. 
\end{cor}