Let $X$ be non-empty set and $S$ be the set of all subsets of $X$, which will be a poset, under subset relation.
Let $\phi \colon S\rightarrow \mathbb{Z}$ be any function and for each $H\in S$, define $\sigma(H)=\sum_{K\leq H}\phi(K)$.
$\mu$ be the Möbius function defined by $\sum_{K\leq H} \mu(H)=\delta_{K,X}$.
How to show that $\sum_{H\leq X} \sigma(H)\mu(H)=\phi(X)$?
(I considered LHS, and substituted for $\sigma(H)$, interchanged the sum, but I couldn't proceed further...)