S}\mid \t\in\Theta\}$ is pointwise bounded and uniformly equicontinuous for $\tilde S=\{z+h_\t(x)\mid z\in C_\tau, \t\in\Theta, x\in[0,1]\}$ with $C_\tau=[c_1-\tau,c_2+\tau]$ (for $C=[c_1,c_2]$ from \[N1\]) for some $\tau>0$, i.e.  $\sup_{\t\in\Theta}|\Lambda_0(\Lambda_\t^{-1}(z))|<\infty$ for all $z\in \tilde S$, and for every $\delta>0$ there exists some $\gamma>0$ such that $\sup_{\t\in\Theta}|\Lambda_0(\Lambda_\t^{-1}(x))-\Lambda_0(\Lambda_\t^{-1}(z))|\leq\delta$ for all $x,z\in \tilde S$ with $|x-z|\leq\gamma$.

5.  \[F2\] For some $\tau>0$, $F_0$ is uniformly continuous on the set $\tilde C=\{\Lambda_0(\Lambda_\t^{-1}(y+a+h_\t(x)))-h_0(x)\mid y\in C, \t\in\Theta,x\in[0,1],|a|\leq \tau\}$ (with $C$ from \[N1\]), i.e. for every $\epsilon>0$ there is some $\delta>0$ such that $|F_0(y)-F_0(z)|<\epsilon$ if $|y-z|\leq\delta$, $y,z\in \tilde C$.

6.  \[smoothing\] $K$ is a density with support $[-1,1]$ and $b_n\searrow 0$, $nb_n\to\infty$.

Let us now make few comments regarding these assumptions.

-   \[M1\] is a common assumption in M-estimation and needed for uniqueness of the true parameter.

-   \[L0\] implies the existence of continuous inverse functions $\Lambda_\t^{-1}$. Further note that uniform equicontinuity and pointwise boundedness imply totally boundedness by the Arzelà-Ascoli theorem. Thus for each $\epsilon$ there is a finite covering of the classes $\mathcal{L}_S$ from \[L1\] and $\mathcal{L}_{\tilde S}^1$ from \[L2\] with balls of radius $\epsilon$ with respect to the sup norm. Thus also the sup norm bracketing numbers of those classes are finite, i.e. $$\begin{aligned}
    \label{bracketing}
    N_{[\,]}(\epsilon,\mathcal{L}_S,\|\cdot\|_\infty)<\infty, \quad N_{[\,]}(\epsilon,\mathcal{L}^1_{\tilde S},\|\cdot\|_\infty)<\infty  