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\documentclass[11pt,a4paper]{article}
\usepackage{blindtext}
\usepackage{amsmath}

\begin{document}
\begin{multline*}
\epsilon(r_1,...,r_n)\\= 
 sgn\prod_{\substack{1 \leq i<j \leq k}}(r_j-r_i)\times
 sgn\prod_{\substack{k+1 \leq i<j \leq n}}(r_i-r_j)\times
 sgn\prod_{\substack{1 \leq i \leq k\\
        k+1\leq j \leq n}}(r_i-r_j).
\end{multline*}

\begin{flalign*}
=&\epsilon(r_1,...,r_k)\times
\epsilon(r_k+1,...,r_n)\times
(-1)^{v_1+...+v_k}
&\end{flalign*}
\end{document}
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2 Answers 2

up vote 5 down vote accepted

You could use a single align* environment to typeset the entire expression.

Some additional comments on your LaTeX code. (i) The \substack instructions after the first two \prod directives don't do anything and should be omitted. (ii) You may wish to encase the subscript terms of the three \prod directives in \mathclap to bring about a nice and tight look of the material on first line. (iii) Don't type ...; instead, type \dots and let LaTeX typeset the typographically appropriate ellipsis. (iv) Since "sgn" would appear to be a math operator, it should probably be typset in a roman (upright) font shape; I suggest declaring a math operator using the \DeclareMathOperator macro.

enter image description here

\documentclass[11pt,a4paper]{article}
\usepackage{mathtools}
\DeclareMathOperator{\sgn}{sgn}
\begin{document}
\begin{align*}
\epsilon(r_1,\dots,r_n)
&=
 \sgn\prod_{\mathclap{1 \leq i<j \leq k}} (r_j-r_i)\times
 \sgn\prod_{\mathclap{k+1 \leq i<j \leq n}} (r_i-r_j)\times
 \sgn\prod_{\mathclap{\substack{1 \leq i \leq k\\ k+1\leq j \leq n}}} (r_i-r_j)\\
&=
 \epsilon(r_1,\dots,r_k)\times \epsilon(r_k+1,\dots,r_n)\times (-1)^{v_1+\dots+v_k}
\end{align*}
\end{document}
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Just to note that mathtools offers the \smashoperator command. –  Manuel Aug 26 at 21:10
1  
@Manuel - That's right. When used in displaymath mode, the outputs of \sgn\prod_{\mathclap{1 \leq i<j \leq k}}(r_j-r_i) and \sgn\smashoperator{\prod_{1 \leq i<j \leq k}}(r_j-r_i) are -- probably not coincidentally... -- identical. –  Mico Aug 26 at 21:15

You can add a white space of the width you need for align both equations in different environments. In this case a\quad produces an em space (a space with the width of a capital em of the font in the size and weight you're using); that seems to be enough for your case.

From your own MWE:

\documentclass[11pt,a4paper]{article}
\usepackage{blindtext}
\usepackage{amsmath}

\begin{document}
\begin{multline*}
\epsilon(r_1,...,r_n)\\= 
 sgn\prod_{\substack{1 \leq i<j \leq k}}(r_j-r_i)\times
 sgn\prod_{\substack{k+1 \leq i<j \leq n}}(r_i-r_j)\times
 sgn\prod_{\substack{1 \leq i \leq k\\
        k+1\leq j \leq n}}(r_i-r_j).
\end{multline*}

\begin{flalign*}
\quad =&\epsilon(r_1,...,r_k)\times
\epsilon(r_k+1,...,r_n)\times
(-1)^{v_1+...+v_k} &
\end{flalign*}
\end{document}

Result

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2  
Avoid using consecutive display math environments. –  egreg Aug 26 at 21:09

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