Let $T$ be a scheme over an algebraically closed field $k$. Let $t \in |T|$ be fixed, and let $i: T \hookrightarrow T'$ be an extension of $T$ by $k(t) := \mathcal{O}_{T,t}/\frak{m}_t$, i.e., $i: T \subset T'$ is a closed immersion with sheaf of ideal $\mathcal{I} \cong k(t)$ such that $\mathcal{I}^2=0$.
Let $T_n := \operatorname{Spec} \mathcal{O}_{T,t}/\frak{m}_t^n$ and let $T_n' := \operatorname{Spec} \mathcal{O}_{T', i^{-1}(t)}/\frak{m}_{i^{-1}(t)}^n$.
Why does there exists some shrinking of $T$ as a neighborhood of $t$, such that $T_n \hookrightarrow T_n'$ is an extension of $T_n$ by $k(t)$?
This claim is made in the proof of Satz (Theorem) 3.2 in H. Flenner's "Ein Kriterium fur die Offenheit der Versalitat": Here is the precise statement:
"Sei $i: T \to T'$ eine Erweiterung von $T$ durch $M=\mathcal{O}(t)=$ und