I am using the textbook Stability, Instability and Chaos by Glenninding to learn dynamical systems. The following excerpt is from the textbook's proof of Theorem 1.14:
(1.14) Theorem: The set $\Lambda(x)$ is invariant, and if $\gamma^{+}(x)$ is bounded then $\Lambda$ is non-empty and compact.
Proof: We begin with the invariance property. Choose $y \in \Lambda(x)$. Then we need to show that $\psi(y, t) \in \Lambda \;\forall \;T$. By the definition of $\Lambda(x) \; \exists$ a sequence $\{t_{i}\}$, with $t_{i} \to \infty$ as $i \to \infty$ s.t. $\psi(x, t_{i}) \to y$ as $i \to \infty$. Now fix $t$ and choose $t_{i}$ sufficiently large so that $t + t_{i} > 0$ then: $$ \psi(x, t + t_{i}) = \psi(\psi(x, t_{i}), t) \to \psi(y, t) \; \textrm{as}\; i \to \infty $$ and so $\psi(y, t) \in \Lambda(x)$. Hence $\Lambda(x)$ is invariant. ...
In the textbook $\Lambda(x)$ is defined to be the $\omega$-limit set and $\gamma^{+}(x)$ is the positive time trajectory. $\psi(x, t)$ is a flow.
Question: How is $\psi(x, t + t_{i}) = \psi(\psi(x, t_{i}), t)$?
In the book this essentially follows from the Existence and Uniqueness theorem in ODE's: if $t\mapsto \psi(x_0,t)$ is the unique solution to $x'=f(x)$ ($f$ at least $C^1$) with initial condition $x_0$, then $t\mapsto \psi(\psi(x_0,s),t)$ is the unique solution to $x'=f(x)$ with initial condition $\psi(x_0,s)$. But then $t\mapsto \psi(x_0,s+t)$ also solves $x'=f(x)$ with initial condition $\psi(x_0,s)$, so
$$\psi(x_0,s+t) = \psi(\psi(x_0,s),t).$$
See also Lem.1.5 on p.16 in the same book.
Alternatively, for topological dynamics one can also take this property to be an axiom.