Tight closure/Relation to Hilbert-Kunz multiplicity/Fact

Let ${}(R,{\mathfrak {m}})$ be an analytically unramified and formally equidimensional local noetherian ring of positive characteristic, let ${}I\subseteq R$ be an ${}{\mathfrak {m}}$-primary ideal. Let ${}f\in R$.

Then

$f\in I^{*}{\text{ if and only if }}e_{HK}((I,f))=e_{HK}(I).$

Write a proof