and the realizing event. (b) The event splits each curve into two subcurves. The hatched areas indicate parts that disappear after the split.[]{data-label="fig:split"}](split)

For induction, we split the two curves based on events (see Fig. \[fig:split\]). Since each split must reduce the problem size, we ignore any events on the left or bottom boundary of cell $(1,1)$ or on the right or top boundary of cell $(m,n)$. This excludes both events of type A. A free space diagram is *connected* at value $\varepsilon$, if a monotonous path exists from the boundary of cell $(1,1)$ to the boundary of cell $(m,n)$. A *realizing event* is a critical event at the minimal value $\varepsilon$ such that the corresponding free space diagram is connected.

Let $\mathcal{E}$ denote the set of concurrent realizing events for two curves. A *realizing set* $E_\text{r}$ is a subset of $\mathcal{E}$ such that the free space admits a monotonous path from cell $(1,1)$ to cell $(m,n)$ without using an event in $\mathcal{E} \backslash E_\text{r}$. Note that a realizing set cannot be empty. When $\mathcal{E}$ contains more than one realizing event, some may be “insignificant”: they are never required to actually make a path in the free space diagram. A 