1$. To make it valid for all $\tau\geq1$ which may be of interest, we should have applied the Main Lemma in Section \ref{NF} not once but twice, thus bringing the remainder down in size to $O(\varepsilon^2)$, which would be enough to apply Theorem \ref{KAM1}. But this would not change almost anything as far as Assumption \ref{hyperbolicity} or the system of equations (\ref{equations}) are concerned, since the added terms would be of higher order in $\varepsilon$. With this final remark we conclude the proof of the Main Theorem. $\Box$ \section*{Appendix A. The Absolute Norm.} We call the norm we use here the {\it absolute norm}. We introduce more notation for an arbitrary function $u(x,\varphi,p,q)$, analytic in the domain $C_{r,s,\kappa}$ for some open set $C\subseteq R^l$. As before $x\in C$ stands for either the action or the frequency variables, which are related through the globally invertible frequency map. For $(x,p,q )\in C_{r,\kappa}$ we set \[ |u|_{x,p,q,s}\equiv \sum_{k\in Z^{n-1}} \overline{u_k(x,p,q)}e^{|k|s}, \] in the notation of Section \ref{not}, \[ |u|_{x,\kappa,s}\equiv \sup_{B^2_\kappa} |u|_{x,p,q,s}. \] \newtheorem{xlemma}{Lemma} \begin{xlemma}[Lemma A1] For the $sup-$norm $|\cdot|^\infty_{r,s,\kappa}$ on $C_{r,s,\kappa}$ for all $\sigma>0,\ \eta>0$ one has: \[ |u|^\infty_{r,s,\kappa}\leq |u|_{r,s,\kappa}\leq {(\kappa+\eta)^2\over\eta^2}\coth^{n-1}\sigma |u|^\infty_{r,s+2\sigma,\kappa+\eta}. \] \end{xlemma} \noindent {\bf Proof.} The first inequality is obvious. To prove the second one let's consider the absolute sum of the Taylor series of a function $u(\cdot,p,q)$ analytic in $p,q$ in $B^2_{\kappa+\eta}$: \[ u(\cdot,p,q)=\sum^\infty_{p,q=0}u_{ij}(\cdot)p^i q^j \] \noindent Clearly for a monomial $u_{ij}p^i q^j$ the coefficient equals: \[ u_{ij}={1\over i!j!}{\partial^{i+j} \over\partial p^i\partial q^j} u(0,0)\leq{1\over (\kappa +\eta)^{i+j}} |u|^\infty_{r,s,\kappa+\eta} \] \noindent here we skip the $(\cdot)$-dependence of $u$, and so \[ \overline{u(p,q)}\leq |u|^\infty_{\kappa+\eta} \left( 1+ {p\over \kappa+\eta} +{p^2\over (\kappa+\eta)^2}+...\right)\cdot \left( 1+{q\over \kappa+\eta} +{q^2\over (\kappa+\eta)^2}+...\right) \] \[ \leq{(\kappa+\eta)^2\over\eta^2} |u|^\infty_{\kappa+\eta}. \] \noindent Then using the familiar estimate for the Fourier coefficients $|u_k(x,p,q)|\leq e^{-|k|s}|u|^\infty_{r,s,\kappa}$ of the analytic functions and the identity $\sum_{k\in Z^{n-1}} e^{-2|k|\sigma}=\coth^{n-1}\sigma$, we come to the claimed result.$\Box$ \begin{xlemma} For an arbitrary $x\in C_r \ |uv|_{x,s,\kappa}\leq |u|_{x,s,\kappa}|v|_{x,s,\kappa}$ for arbitrary functions $u,v$, analytic in $C_{r,s,\kappa}$. \end{xlemma} \noindent {\bf Proof.} For any $(x,p,q)\in C_{r,s,\kappa}$ we have \[ |uv|_{x,p,q,s}=\sum_k \overline{(uv)_k (x,p,q)}e^{|k|s} \leq \sum_k \sum_l \overline{u_{k-l} (x,p,q)}e^{|k-l|s} \overline{v_l (x,p,q)}e^{|l|s} \] \[ =|u|_{x,p,q,s}|v|_{x,p,q,s}. \Box \] \begin{xlemma} For $0<\rho