Kato’s monograph [@Kato]). It was shown in [@Nicolaescu] (see also [@Lesch Prop. 2.2]) that every path of selfadjoint Fredholm operators that is mapped to a continuous path of bounded operators under is also continuous with respect to the gap-metric. Finally, Booss-Bavnbek, Lesch and Phillips constructed in [@UnbSpecFlow] the spectral flow for paths of selfadjoint Fredholm operators in this more general setting. The main result of this paper now reads as follows (see [@Cappell Thm. 0.4]).

\[main\] If $(\gamma_1,\gamma_2)$ is a pair of paths in $\Lambda(n)$, then the family of differential operators is continuous with respect to the gap-metric and

$$\operatorname{sf}(\mathcal{A})=\mu_{Mas}(\gamma_1,\gamma_2).$$

Let us make a few comments on our proof. Firstly, we want to emphasise that we prove the gap-continuity of the family from first principles just by elementary estimates and standard facts about orthogonal projections that can all be found in the monograph [@Kato]. Secondly, our proof of the spectral flow formula in Theorem \[main\] is surprisingly simple. We assume at first that

$$\begin{aligned}
\label{introduction-transversal}
\gamma_1(0)\cap\gamma_2(0)=\gamma_1(1)\cap\gamma_2(1)=\{0\}\end{aligned}$$

and show that the Maslov index can be characterised in this case by three axioms. This uniqueness theorem needs nothing else than the elementary properties of the Maslov index and the fact that the 