Rewrite relations - proof of correctness
To simplify, let $D$ be the domain of $T$ and let $R = \{\epsilon\} \cup (\Sigma^* \setminus \Sigma^*D\Sigma^*)$. Then by definition
$$
N(T) = Id_R \quad \text{and} \quad R^{obl}(T) = N(T)(TN(T))^*.
$$
Here is a formal way to justify your idea.
Let $(u,v) \in \Sigma^* \times \Sigma^*$. By definition, $(u,v) \in R^{obl}(T)$ if and only if $(u,v)$ can be written as
$$
(u, v) = (r_0, r_0)(u_1, v_1)(r_1, r_1)(u_2,v_2) \dotsm  (r_{k-1}, r_{k-1})(u_k,v_k)(r_k, r_k)
$$
where $r_0, r_1, \ldots, r_k \in R$ and $(u_1, v_1), \ldots, (u_k,v_k) \in T$, which is exactly what you describe in your sentence starting by &quot;The idea is the following&quot;.