Poset generated by intervals with inclusion

Question

Let $P$ be a poset and consider the poset $Q$ generated by all closed intervals of $[x,y]\subseteq P$ with inclusion, that is $[x,y]\leq_{Q} [z,w] \iff x\geq_Pz$ & $y\leq_P w$. We add a minimum element $\hat{0}$ and a maximum element $\hat{1}$ to $P,Q$. Show that $\mu_P(\hat{0},\hat{1})=\mu_Q(\hat{0},\hat{1})$, where $\mu_P$ is the Mobius function of $P$.