Let $u,v$ be random variables with $u$ being related to $v$.
The expression is $\mathbb{E}\left[ \mathbb{E}\left(\left(u - \mathbb{E}(u|v)\right)^{2} \right)\right | v]$.
Expanding the expression in the second outer nested expectation $\mathbb{E}$ seems trivial:
$(\left(u - \mathbb{E}(u|v)\right)^{2} = u^{2} - 2u\mathbb{E}(\left(u|v\right) + \left(\mathbb{E}\left(u|v\right)\right)^{2}$
$\rightarrow \mathbb{E}\left(\left(u - \mathbb{E}(u|v)\right)^{2} |v\right)$ = $\mathbb{E}\left(u^{2}|v\right) -2\left(u \mathbb{E}\left(u|v\right)|v\right) + \mathbb{E}\left( \left(\mathbb{E}\left(u|v\right)\right)^{2} | v\right)$
I'm stuck at this point and would welcome some hints or identities to get me through.