Let $X,Y$ be algebraic subsets of $\mathbb A^n.$ I would like to show that if $X$ and $Y$ intersect transversely (i.e. $T_p\, X\cup T_p\, Y=T_p\, \mathbb A^n$ for every $p\in X\cap Y$) then $I(X)+I(Y)$ is radical (so $I(X\cap Y)=I(X)+I(Y)$). How to prove it?

It is possible that my definition of transversality needs to be strengthened. This statement was conjectured in this math stackexchange answer (see the top answer), but not proved, so I am hoping to find a proof here.