Local ring of product of affine varieties at a point

Question

Let $X \subseteq \mathbb{A}^n, Y\subseteq \mathbb{A}^m$ be closed algebraic sets over an (algebraically closed) field $k$. Let $x \in X$, $y \in Y$, $\mathcal{O}_{X,x}$ the local ring of $X$ at $x$ (the localization of $k[X]$ at the maximal ideal corresponding to $x$). Let $M=\mathcal{O}_{X,x} \otimes m_y+m_x \otimes \mathcal{O}_{Y,y}$, where $m_x$ and $m_y$ are the maximal ideals of $\mathcal{O}_{X,x}$ and $\mathcal{O}_{Y,y}$. I would like a proof of the $k$-algebra isomorphism \begin{align} \mathcal{O}_{X\times Y,(x,y)}\cong (\mathcal{O}_{X,x} \otimes \mathcal{O}_{Y,y})_M, \end{align} where the righthand side denotes the localization of $\mathcal{O}_{X,x} \otimes \mathcal{O}_{Y,y}$ at $M$. You may freely use the fact that $k[X \times Y]\cong k[X]\otimes_k k[Y]$.

Answer 1

Thank you Armando j18eos for the comment that this proof can be found here: tag 01jO.