correct matching $\mu_i = (\sigma_i,\theta_i)$ exists for $P_i$ and $Q_i$ such that $\mu_i$ is linear in every cell and $d_{\mu_i}(t) \leq \varepsilon_\text{r}$ for all $t$ with $\sigma_i(t) \geq 1$ or $\theta_i(t) \geq 1$, and $\sigma_i(t) \leq m_i-1$ or $\theta_i(t) \leq n_i -1$. Combining these matchings with the events in $E_\text{r}$ yields a matching $\mu = (\sigma,\theta)$ for $(P,Q)$.

As we argue below, this matching is locally correct and satisfies the additional properties. The matching of an event corresponds to a single point (type B) or a horizontal or vertical line (type C). By induction, $\mu_i$ is linear in every cell. Since all events occur on cell boundaries, the cells of the matchings and events are disjoint. Therefore, the matching $\mu$ is also linear inside every cell.

For $i < k+1$, $d_{\mu_i}$ is at most $\varepsilon_\text{r}$ at the point where $\mu_i$ enters cell $(m_i,n_i)$ in the free space diagram of $P_i$ and $Q_i$. We also know that $d_{\mu_i}$ equals $\varepsilon_\text{r}$ at the top right corner of cell $(m_i,n_i)$. Since $\mu_i$ is linear inside the cell, $d_{\mu_i}(t) \leq \varepsilon_\text{r}$ also holds for $t$ with $\sigma_i(t) > m_i-1$ and $\theta_i(t) > n_i -1$. Analogously, for $i > 0$, $d_{\mu_i}(t)$ is at most $\varepsilon_\text{r}$ 