%% 
%%  An UIT Edition example
%% 
%%  Example 10-06-2 on page 209.
%% 
%%  Copyright (C) 2010 Herbert Voss
%% 
%%  It may be distributed and/or modified under the conditions
%%  of the LaTeX Project Public License, either version 1.3
%%  of this license or (at your option) any later version.
%% 
%%  See http://www.latex-project.org/lppl.txt for details.
%% 
%% 
%% ==== 
% Show page(s) 1
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\documentclass[]{exaarticle}
\pagestyle{empty}
\setlength\textwidth{375.57637pt}
\AtBeginDocument{\setlength\parindent{0pt}}
\StartShownPreambleCommands
\usepackage{amsmath}
\usepackage[sans]{dsfont}
\StopShownPreambleCommands
\begin{document}
\begin{align*}
  \sum_{i=n}^m a_i &= \sum_{j=n}^m a_j \tag{index renaming}\\
  \sum_{i=n}^m a_i &= \sum_{i=n-l}^{m-l} a_{i+l} &\forall\, l\in\mathds{Z}{}
	 \tag{index shift},\\
  \sum_{i=n}^m a_i+\sum_{i=m+1}^k a_i &= \sum_{i=n}^k a_i,
                            &\text{if } n\le m<k,\\
  \sum_{i=n}^m a_i+\sum_{i=n}^m b_i &= \sum_{i=n}^m (a_i+b_i),\\
  \sum_{i=n}^m c\times a_i &= c\times\sum_{i=n}^m a_i & \forall\, c\in\mathds{R}{,}\\
  \sum_{i=n}^m a_i \times\sum_{j=l}^k b_j &=
                            \sum_{i=n}^m\sum_{j=l}^k a_i b_j
\end{align*}
\end{document}
