and $\mu_{Mas}$ are indeed equal for closed paths based at $(L_0,L_1)$.\
Let us now assume that $(\gamma_1,\gamma_2)\in\Omega^2$ is an arbitrary admissible pair of paths. We connect $(L_0,L_1)$ to $(\gamma_1(0),\gamma_2(0))$ by a pair of paths $(\gamma_3,\gamma_4)$ and $(\gamma_1(1),\gamma_2(1))$ to $(L_0,L_1)$ by a pair of paths $(\gamma_5,\gamma_6)$, where we can assume by Lemma \[lemma-connected\] that $\gamma_3(\lambda)\cap\gamma_4(\lambda)=\gamma_5(\lambda)\cap\gamma_6(\lambda)=\{0\}$ for all $\lambda$. Then by (i’), (ii’) and the first step of our proof

$$\begin{aligned}
\mu(\gamma_1,\gamma_2)&=\mu(\gamma_3,\gamma_4)+\mu(\gamma_1,\gamma_2)+\mu(\gamma_5,\gamma_6)\\
&=\mu((\gamma_3,\gamma_4)\ast(\gamma_1,\gamma_2)\ast(\gamma_5,\gamma_6))=\mu_{Mas}((\gamma_3,\gamma_4)\ast(\gamma_1,\gamma_2)\ast(\gamma_5,\gamma_6))\\
&=\mu_{Mas}(\gamma_3,\gamma_4)+\mu_{Mas}(\gamma_1,\gamma_2)+\mu_{Mas}(\gamma_5,\gamma_6)=\mu_{Mas}(\gamma_1,\gamma_2),\end{aligned}$$

which proves the proposition.

Let $(\gamma_1,\gamma_2)$ be a pair of paths in $\Lambda(n)$ as in (i’), i.e. $\gamma_1(\lambda)\cap\gamma_2(\lambda)=\{0\}$ for all $\lambda\in I$. Then $(\gamma_1,\gamma_2)$ is homotopic to the constant pair of paths $(\widetilde{\gamma}_1(\lambda),\widetilde{\gamma}_2(\lambda))=(\gamma_1(0),\gamma_2(0))$, $\lambda\in I$, by a homotopy of admissible pairs. Hence $\mu(\gamma_1,\gamma_2)=\mu(\widetilde{\gamma}_1,\widetilde{\gamma}_2)$ by (iii’). As

$$\mu(\widetilde{\gamma}_1,\widetilde{\gamma}_2)=\mu((\widetilde{\gamma}_1,\widetilde{\gamma}_2)\ast(\widetilde{\gamma}_1,\widetilde{\gamma}_2))=\mu(\widetilde{\gamma}_1,\widetilde{\gamma}_2)+\mu(\widetilde{\gamma}_1,\widetilde{\gamma}_2)$$ by (ii’), we see that $\mu(\gamma_1,\gamma_2)=\mu(\widetilde{\gamma}_1,\widetilde{\gamma}_2)=0$ and so (i’) follows from (ii’) and (iii’). Consequently, Proposition \[prop-Maslov\] actually characterises the Maslov index by the three axioms (ii’), (iii’) and (N).

We now define

$$\mu:\Omega^2\rightarrow\mathbb{Z},\quad \mu(\gamma_1,\gamma_2)=\operatorname{sf}(\mathcal{A}),$$ where $\mathcal{A}$ is the path of differential operators for the pair $(\gamma_1,\gamma_2)$. We aim to use Proposition \[prop-Maslov\] to show Theorem \[main\] and so we need to check the properties (i’), (ii’), (iii’) and (N). Let us first note that (i’) follows immediately from (ii) in 