u$ is a maximizer in $S(t_0+\tau, u(t_0 + \tau; t_0, a))$. With similar modifications, following the proof of Theorem \[thm:main\], one verifies property [**LS4**]{} for $V_\zeta[S]$. Thus, for any functional $\zeta$ of the form , $V_\zeta[S(t_0, a)]$ is a family of abstract local integral funnels.

Choose a countable family $\Phi$ of continuous functions $\vp : X\to [0, 1]$ that separates the points of $X$, and 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 $S^0(t_0, a) = S(t_0, a)$, and $S^{n}(t_0, a) = V_{\zeta_n}[S^{n-1}(t_0, a)], n = 1, 2, \dots $. Again, the intersection, $$S^\infty(t_0, a) = \bigcap_{n = 0}^\infty S^{n}(t_0, a)\,,$$ is an abstract family of local integral funnels. To show that each funnel $S^\infty(t_0, a)$ is a singleton, assume $u, v\in S^\infty(t_0, a)$ Then, for every $\vp\in \Phi$, $$\int_0^{T(t_0, a)}  e^{-\lambda t} \vp(u(t_0 + t; t_0, a))\,dt = \int_0^{T(t_0, a)} e^{-\lambda t} \vp(v(t_0 + t; t_0, a))\,dt\,,\quad \forall \lambda\in {{\mathbb{Q}}}_+\,.$$ By the uniqueness of the Laplace transform, $\vp(u(t_0 + t; t_0, a)) 