Let $X \in \mathcal{L}^2(\Omega,\mathcal{F},\mathbb{P})$ and $\mathcal{A} \subset \mathcal{F}$ be a sub-$\sigma$-algebra. From the definition of the conditional expectation it follows that $\mathbb{E}[X]=\mathbb{E}[\mathbb{E}[X\;| \mathcal{A}]]$. But why do we obtain that even
$$\mathbb{E}[\;X \;\mathbb{E}[X| \mathcal{A}]\;]=\mathbb{E}[\;\mathbb{E}[X\;| \mathcal{A}]^2\;]$$ holds?