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Consider the set of all subsets of a set S.  A chain is a collection of subsets $P_1 \subset P_2 \subset P_3 \subset P_4 \dots  \subset P_k$.  A symmetric chain is one which starts at a set of size $i$ and ends at a set of size $n - i$.  Prove that the poset has a decomposition into symmetric chains.

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