Let the function $R(x_1, x_2) \,= \, (R_1(x_1, x_2),\, R_2( x_1, x_2))$ be defined by $R_i(x_1, x_2) = \\=  x_i r_{x_i}(1, x_1, x_2)/r(1, x_1, x_2)$ for $i=1, 2,$ $r_{x_i}(1, x_1, x_2)$ is the partial derivative of $r(1, x_1, x_2)$ wrt. $x_i.$ The Jacobi Determinant of $R(x_1, x_2) $ is $>0$ at $ x_1=x_2=1$ (proof Subsection \[LOLICO\].) Thus there is a neighborhood of $(1, 1)$ in which $R( x_1, x_2)$ is invertible and the inverse function is differentiable. We have that $R(1, 1)= (k/3, k/3).$ Thus for a suitable $\varepsilon $ and $(\lambda_1, \lambda_2) \in  {\cal U}_\varepsilon(1/3, 1/3)$ we can define $(c_1, c_2)$ by $R(c_1, c_2)= (k \lambda_1, k\lambda_2).$ Moreover, $c_i=c_i(\lambda_1, \lambda_2)$ is differentiable. Lemma \[KLOC\] is proved in Subsection \[LOLICO\].

\[KLOC\] There is an $\varepsilon>0$ such that for $(\lambda_1, \lambda_2) \in  {\cal U}_\varepsilon(1/3, 1/3)$ $$\begin{aligned}
K(\bar{l}) \,= \, O\left( \frac{1}{n}\right) \cdot \frac{r(1, c_1, c_2)}{c_1^{l_1}c_2^{l_2}} \mbox{ with } R(c_1, c_2)=(k\lambda_1, k\lambda_2) \mbox{ defining } c_1, c_2.  \nonumber \end{aligned}$$

\[PSIINU\] There is $\varepsilon >0$ such that for $(\omega_1, \omega_2), ( \lambda_1, \lambda_2 ) \in   {\cal U}_\varepsilon(1/3, 1/3)$ $$\begin{aligned}
\frac{N(\bar{w}, \bar{l})}{N_0} \, \, \le  \, 
\,O \left(\frac{1}{n^2}\right) \Psi( \bar{\omega}\, ,\, \bar{\lambda}\, ,\, \bar{a} \,,\, \bar{c}  \, )^n \nonumber \end{aligned}$$ where $  Q(a_i)= l_i 