of (\[inequality\_for\_contraction\]) gives $$\label{zwischen4}
\int_{\Gamma_{\bar{t}_1}^{\bar{t}}}\partial_{x}\xi_{\varepsilon}(x,t)\,dx\,dt=\int_{t_{1}}^{\bar{t}}\xi_{\varepsilon}(\theta_{\varepsilon}^{+}(t),t)-\xi_{\varepsilon}(\theta_{\varepsilon}^{-}(t),t)\,dt\,.$$ Inserting (\[zwischen3\]) and (\[zwischen4\]) back in (\[inequality\_for\_contraction\]) leads to the identity $$\label{main_equality_lemma}\begin{split}
\int_{\theta_{\varepsilon}^{-}(\bar{t})}^{\theta^{+}_{\varepsilon}(\bar{t})}\eta_{\varepsilon}(x,\bar{t})\,dx&=\int_{\Lambda_{\bar{t}_1}^{\bar{t}}}\begin{pmatrix}\eta_{\varepsilon}\\-\xi_{\varepsilon}\end{pmatrix}\cdot\tau\,d\sigma- \int_{\Gamma_{\bar{t}_1}^{\bar{t}}}\int_{\mathbb{R}}\eta''(a)q(x,t,a)\,da\,.
\end{split}$$ For suitable choices of $\eta$ this equality (\[main\_equality\_lemma\]) will imply the first two claims of Lemma \[properties\_of\_entropy\_solutions\].

First we prove (\[Linfty\_bound\]). Let $a_0$ be a real number being fixed later in this proof. We choose $$\eta(a)=\left\{\begin{array}{cl}
                (a-a_{0})&\mbox{if}~~a-a_{0}\geq0\,,\\
        0 &\mbox{if}~~a-a_{0}\leq0\
               \end{array}\right.$$ and we aim to deduce $$\label{positive_part_contraction}
 \int_{\theta^{-}(\bar{t})}^{\theta^{+}(\bar{t})}|v(x,\bar{t})-a_{0}|^{+}\,d\sigma\leq C \int_{\Lambda_{t_{1}}^{\bar{t}}}|v_{0}(x)-a_{0}|^{+}\,d\sigma\,,$$ from equality (\[main\_equality\_lemma\]). The non-negativity of $\eta''(a)$ and $q_{\varepsilon}$ implies $$\int_{\Gamma_{\bar{t}_1}^{\bar{t}}}\int_{\mathbb{R}}\eta''(a)q(x,t,a)\,da\ge 0\quad .$$ Using this inequality in equality (\[main\_equality\_lemma\]), we obtain the estimate $$\int_{\theta_{\varepsilon}^{-}(\bar{t})}^{\theta^{+}_{\varepsilon}(\bar{t})}\eta_{\varepsilon}(x,\bar{t})\,dx\leq\int_{\Lambda_{\bar{t}_1}^{\bar{t}}}\begin{pmatrix}\eta_{\varepsilon}\\-\xi_{\varepsilon}\end{pmatrix}\cdot\tau\,d\sigma\,.$$ Letting $\varepsilon\to0^+$ we get $$\int_{\theta^{-}(\bar{t})}^{\theta^{+}(\bar{t})}\eta(x,\bar{t})\,dx\leq\int_{\Lambda_{t_1}^{\bar{t}}}\begin{pmatrix}\eta(v_1)\\-\xi(v_1)\end{pmatrix}\cdot\tau\,d\sigma\,.$$ We observe $$\label{bound_for_xi}
 \left|\xi(a)\right|\leq \max_{|b|\leq\|v_1\|_{\infty}}|f'(b)|\cdot\eta(a)\,,$$ which implies $$\int_{\theta^{-}(\bar{t})}^{\theta^{+}(\bar{t})}\eta(x,\bar{t})\,dx\leq C\int_{\Lambda_{t_1}^{\bar{t}}}\eta(v_1)\,dx\,.$$ This is our desired result (\[positive\_part\_contraction\]) and choosing $a_0=\|v_1\|_{\infty}$ in (\[positive\_part\_contraction\]) gives $$\int_{\theta_{-}(\bar{t})}^{\theta^{+}(\bar{t})}|v(x,\bar{t})-a_{0}|^{+}\,d\sigma=0$$ and thus (\[Linfty\_bound\]) follows: $$|v(x,t)|\leq \|v_1\|_{\infty}\quad\mbox{a.e. in}\quad\Gamma_{t_1}^{t_2}\,.$$

In order to prove (\[control\_of\_the\_defect\_measure\]), we choose now $$\eta(a):=\left\{\begin{array}{ccl}
                  2a^{2}&\mbox{if}&a\geq-\|v\|_{\infty}\,,\\
          (a+\|v\|_{\infty})+2\|v\|_{\infty}^{2}&\mbox{if}&-(\|v\|_{\infty}+2\|v\|_{\infty}^{2})\leq a\leq-\|v\|_{\infty}\,,\\
          0 &\mbox{if}&a\leq-(\|v\|_{\infty}+2\|v\|_{\infty}^{2})\,.
                 \end{array}\right.$$ Since $\eta$ is non-negative, we deduce from (\[main\_equality\_lemma\]) $$\label{last_equation_for_mass_bound}\begin{split}
 2\int_{\Gamma_{t_{1}+C\varepsilon}^{t_{2}-C\varepsilon}}\int_{\mathbb{R}}q_{\varepsilon}(x,t,a)\,da\,dx\,dt&\leq \int_{\Lambda_{\bar{t}_1}^{\bar{t}}}\begin{pmatrix}\eta_{\varepsilon}\\-\xi_{\varepsilon}\end{pmatrix}\cdot\tau\,d\sigma\quad .
\end{split}$$ Since $\eta(a)=2a^2$ for $a\in[-\|v\|_{\infty},\|v\|_{\infty}]$ we get $$|\xi(a)|=\int_{-\|v\|_{\infty}}^{a}|\eta'(b)f'(b)|\,db
\leq f'(\|v\|_{\infty}) \int_{-\|v\|_{\infty}}^{|a|}|\eta'(b)|\,db\,.$$ Hence, by letting $\varepsilon\to0$ in (\[last\_equation\_for\_mass\_bound\]), we obain $$\int_{\Gamma_{t_{1}}^{t_{2}}}\int_{\mathbb{R}}q(x,t,a)\,da\,dx\,dt\leq C(\delta+t_2-t_1)\,,$$ as announced in (\[control\_of\_the\_defect\_measure\]).

Finally we are going to prove (\[contraction\_inequality\]). We choose the domain $\Gamma_{t_1}^{t_2}$ in such a way that $$0<\hat{\lambda}\leq \lambda_1\,,$$ where $$\label{def_lambda1}\begin{array}{c}
