
\begin{proof}
  Write \(\mathbf{S} = (S_1, \dots, S_k)\) such that \(\mathbf{S} = (f_{S_1}(\mathbf{X}_{pa(S_1)}, \mathbf{N}_{S_1}), \ldots, f_{S_k}(\mathbf{X}_{pa(S_k)}, \mathbf{N}_{S_k}) )\). It takes finitely many steps to recursively substitute the parents of \(S_i, i \in [k]\) by the corresponding structural equation such that \(\mathbf{S} = f(\mathbf{N}_{A_1}, \ldots, \mathbf{N}_{A_l})\) with \(\{A_1, \ldots, A_l\}\) the set of all ancestors of nodes in \(\mathbf{S}\) that do not contain the node \(g\). The statement follows from the joint independence of the noise variables across groups.
\end{proof}
