account what we have just shown, that $\zeta(\sigma_\tau u) = m_\zeta(t_0 + \tau, u(t_0 + \tau))$. If $v\in V_\zeta[S(t_0+\tau, u(t_0+\tau))]$, then $\zeta(v) = m_\zeta(t_0 + \tau, u(t_0 + \tau))$ as well. Thus, $$\begin{aligned}
& \int_0^\infty e^{-\lambda t} \vp(w(t_0 + t))\,dt = \int_0^\tau e^{-\lambda t} \vp(u(t_0 + t))\,dt + 
\int_\tau^\infty e^{-\lambda t} \vp(v(t_0 + t))\,dt = \\ 
& \int_0^\tau e^{-\lambda t} \vp(u(t_0 + t))\,dt + e^{-\lambda \tau} \zeta(v) = 
\int_0^\tau e^{-\lambda t} \vp(u(t_0 + t))\,dt + e^{-\lambda \tau} \zeta(\sigma_\tau u) = \zeta(u)\,,
\end{aligned}$$ i.e., $\zeta(w) = m_\zeta(t_0, a)$, i.e., $w\in V_\zeta[S(t_0, a)]$. To summarize, for any functional $\zeta$ of the form , $V_\zeta[S(t_0, a)]$ is a family of abstract integral funnels.

Now, choose a countable family $\Phi$ of continuous functions $\vp : X\to [0, 1]$ that separates the points of $X$, see [@B-K]. Choose some enumeration $(\lambda_n, \vp_n)$ of the countable set of pairs $(\lambda, \vp)$, where $\lambda$ runs through positive rational numbers and $\vp$ runs through $\Phi$. To each pair $(\lambda_n, \vp_n)$ corresponds the functional $\zeta_n$ via . Define recursively the shrinking families of abstract integral funnels $$S^0(t_0, a) = S(t_0, a), \quad S^{n}(t_0, a) = V_{\zeta_n}[S^{n-1}(t_0, a)], n = 1, 2, \dots\,.$$ For each $(t_0, a)$, $S^{n}(t_0, a)$ is 