Formal curves are set theoretic complete intersections

Let $k$ be any field. Let $R=k[[X_1,\dots ,X_d]]$ and $\mathfrak{p}$ a $1$-dimensional prime ideal. In this note we present $d-1$ elements such as $f_1,\dots ,f_{d-1}$ in $\mathfrak{p}$ such that $\mathfrak{p}=\text{rad}(f_1,\dots ,f_{d-1})$.