Definition about tensor product of distributions:
Let $u_j \in \mathcal{D}'(\Omega_j)$, $j=1,2$. There is only $u \in \mathcal{D}'(\Omega_1 \times \Omega_2)$ such that $$\langle u, \varphi_1 \otimes \varphi_2 \rangle=\langle u_1, \varphi_1 \rangle \langle u_2, \varphi_2 \rangle$$ $\forall \varphi_j \in C^{\infty}_{c}(\Omega_j).$ Furthermore, if $\varphi \in C^{\infty}_{c}(\Omega)$ then \begin{align} \langle u, \varphi \rangle&=\langle u_1, \langle u_2, \varphi(x_1,x_2)\rangle \rangle \\ &=\langle u_2, \langle u_1, \varphi(x_1,x_2) \rangle \rangle. \end{align}
My problem is
If $u \in \mathcal{D}'(\mathbb{R}^{n})$ and $x_nu=0$ then there is $v\in \mathcal{D}'(\mathbb{R}^{n-1})$ such that $$u(x)=v(x') \otimes \delta(x_n),$$ where $x=(x',x_n)$.
I think I should use the uniqueness of the definition about tensor product.
Thank you for your help.