Let $L$ be a finite lattice with least and greatest elements $0, 1$, respectively, and let $X\subseteq L$. Let $n_k$ be the number of $k$-element subsets of $X$ with join $1$ and meet $0$. I want to show that $$\sum_k (-1)^kn_k=-\mu(0, 1)+\sum_{x\leq y, [x, y]\cap X=\varnothing}\mu(0, x)\mu(y, 1).$$
This result seem to be similar to and an extension of the Crosscut Theorem, but I'm not sure how to prove this result. I also want to possibly use this result to prove the Complementation Theorem:
$$\mu(0, 1)=\sum_{x\leq y}\mu(0, x)\mu(y, 1),$$ where $x, y$ range over pairs such that, for a fixed $t\in L$, $x\vee t=y\vee t=1, x\wedge t=y\wedge t=0$. My idea for this is to take $X$ to be the set of non-complements of $t$, which reduces the problem to showing that $\sum_k (-1)^kn_k=0$ in this case, but I'm not sure how to do that either (or even if my idea is correct).