  We expand the ranges of permissible values of $n$ ($n\in\mathbb{N}$) for
which Poisson kernels
$P_{q,\beta}(t)=\sum\limits_{k=1}^{\infty}q^k\cos\left(kt-\dfrac{\beta\pi}{2}\right)$,
${q\in(0,1)}$, $\beta\in\mathbb{R}$, satisfy Kushpel's condition $C_{y,2n}$. As
a consequence, we obtain exact values for Kolmogorov widths in the space $C$
($L$) of classes $C_{\beta,\infty}^q$ ($C_{\beta,1}^q$) of Poisson integrals
generated by kernels $P_{q,\beta}(t)$ in new situations. It is shown that
obtained here results we can't obtain by using methods of finding of exact
lower bounds for widths suggested by A. Pinkus.
