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\begin{document}
\section*{LaTeX 公式编辑与导出练习}

这是一份自编样例，用于核对公式源码、预览和 Word 成品。

\section*{1. 行内上下标}

勾股关系: \({a}_1^2 + {b}_1^2 = {c}_1^2\) 。

\section*{2. 分式与求和}

\[
y = \frac{x + 1}{x - 1},\;x \neq  1
\]

另起一段，核对求和等式:

\[
\mathop{\sum }\limits_{k = 1}^{n}k = \frac{n\left( {n + 1}\right) }{2}
\]

\section*{3. 二阶矩阵}

\[
A = \left\lbrack  \begin{array}{ll} 1 & 2 \\  3 & 4 \end{array}\right\rbrack
\]

\section*{4. 分段函数}

\[
\left| x\right|  = \left\{  \begin{matrix} x, & x \geq  0 \\   - x, & x < 0 \end{matrix}\right.
\]

\section*{5. 多行等式}

\[
\begin{aligned} f\left( x\right) &  = {\left( x + 1\right) }^2 \\   &  = {x}^2 + {2x} + 1 \end{aligned}
\]

\section*{6. 代码示例保持原样}

\section*{下面一行用于展示源码，不应被渲染为公式:}

\textbackslash frac\{x+1\}\{x-1\}

结束核对:DOC2X-LATEX-EDITOR-END。
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