I have made a shortcut on proving the Mobius Inversion Theorem, which states the following:
Let $N_{e}(x)$ to be a real-valued function, defined for all $x$ in a locally finite partially ordered set $(S, \leq)$ and assume there is an element $m \in S$ such that $N_{e}(x)=0$ where $x \nleq m$. Define $N_{a}(x)$ by $N_{a}(x)= \sum_{y:y \geq x}N_{e}(y)$, then $N_{e}(x)= \sum_{y:y \geq x} \mu(x,y) N_{a}(y)$.
Proof:
Note that $N_{a}(x)= \sum_{y:y \geq x}N_{e}(y)= \sum_{x \leq y\leq m}N_{e}(y)$.
Next, $\sum_{y:y \geq x}N_{a}(y) \mu(x,y)$
\begin{align*}
&=\sum_{x \leq y} \sum_{y \leq z \leq m}N_{e}(z) \mu (x,y)\\
& =\sum_{x \leq y} \sum_{z} N_{e}(x) \zeta (y,z) \mu(x,y) \\
& =\sum_{z}N_{e}(z) \sum_{x \leq y} \mu(x,y) \zeta (y,z) \\
& = \sum_{z}N_{e}(z) \delta (x,z) \\
& =N_{e}(x)
\end{align*}
I don't know what do I get in proving this theorem. From my further reading, $N_{e}(b)$ represents the number of colorings on the vertices of $G$ that have exactly $b$ as their bond. While, the function $N_{a}(b)$ represents the number of colorings on the vertices of $G$ that have at least $b$ as their bond.