Proof. Let $\lambda \in (E \wh \otimes_{\eps} F)^{*}$, then there exists equicontinuous subsets $S \subset E^{*}$ and $T \subset F^{*}$ such that for each $x \in E$ and $y \in F$,

For any $(x, y) \in E \times F$ and $(\phi, \psi) \in S \times T$, let $f_{xy}(\phi, \psi) = \dpn{x, \phi}{E}\dpn{y, \psi}{F}$. By the Hahn-Banach Theorem, there exists $\Lambda \in C(S \times T; K)^{*}$ such that the following diagram commutes:

Now, since $S$ and $T$ are equicontinuous, using the Banach-Alaoglu Theorem, assume without loss of generality that $S$ and $T$ are weak*-compact. In which case, by the Riesz Representation Theorem, there exists $\mu \in M_{R}(S \times T; K)$ such that $\Lambda(f) = \int_{S \times T}f d\mu$ for all $f \in C(S \times T; K)$. Therefore

for all $(x, y) \in E \times F$, and $\lambda \in I(E; F)$.$\square$