I am reading the book "Elements of information theory" by thomas M. Cover and Joy A. Thomas, second edition. In page 255 of the book there is a theorem:

For any random variables $X$ and estimator $\hat{X}$, $E(X - \hat{X})^2 \geq \frac{1}{2\pi e}e^{2h(X)}$. The proof uses at its core the fact that $var(X) \geq \frac{1}{2\pi e}e^{2h(X)}$.

Edit: I include here the proof:

$E(X-\hat{X})^2 \geq \min_{\hat{X}}E(X-\hat{X})^2 = E(X-E(X))^2 = var(X) \geq \frac{1}{2\pi e}e^{2h(X)}$

Right after the proof there is a corollary that says: Given side information $Y$ and estimator $\hat{X}(Y)$, it follows that: $E(X - \hat{X}(Y))^2 \geq \frac{1}{2\pi e}e^{2h(X|Y)}$, however, no further explanation is supplied for how to derive this corollary from the previous theorem.

I assume that if I fight the equations for a while I might be able to derive the corollary from something, however I can't manage to find a simple way to derive it from the previous theorem. I wonder if I miss something very simple here.

So far my best attempt was to split $(X - \hat{X}(Y))$ into $(X - Y) + (Y - \hat{X}(Y))$.