Let $X$ be a set. Suppose that $\sigma_1, \sigma_2, \ldots,\sigma_k$ are all permutations of $X$. Then the composition $\sigma_1\circ \sigma_2\circ \ldots\circ \sigma_k$ is a permutation of $X$.

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theorem.the_composition_of_permutations_is_a_permutation.tex

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\begin{theorem}
Let $X$ be a set. Suppose that $\sigma_1, \sigma_2, \ldots,\sigma_k$ are all [[Definition.Permutation|permutations]] of $X$.  Then the composition 
$\sigma_1\circ \sigma_2\circ \ldots\circ \sigma_k$
is a permutation of $X$.
\end{theorem}

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theorem