I am not sure how to show the following property of the Maslov Index, which in McDuff and Salamon's Introduction to Symplectic Topology, Theorem 2.35, is called the product property.
Let $\Lambda: \mathbb{R}/\mathbb{Z} \rightarrow \mathcal{L}(n)$ and $\Psi :\mathbb{R}/\mathbb{Z} \rightarrow \operatorname{Sp}(2n)$ be two loops, then Maslov index, $\mu$ satisfies: $$\mu(\Psi \Lambda)= \mu(\Lambda) +2\mu(\Psi)$$
It should be easy but I don't see it, they say that it's implied from the Homotopy property of Maslov index, but I am clueless here.
Any hints?
Any input is more than welcome. Thanks.