How to prove $E[UE(V|W)]=E[E(U|W)V]=E[E(U|W)E(V|W)]$?

Question

Suppose $U$, $V$, and $W$ are random variables, random vectors, or random matrices, and the dimensions of $U$ and $V$ are such that the product $UV$ is defined, and suppose the conditional and unconditional expectations involved are defined. Then

$$E[UE(V|W)]=E[E(U|W)V]=E[E(U|W)E(V|W)].$$

How to prove the above relationship?

By following Kroki's hint, I find $E[UE(V|W)]=E(E(UE(V|W)|W))=E(E(V|W)E(U|W))$. Then this is the second equality sign. How to prove the first equality sign?

Answer 1

Hint: $\mathbb E [\mathbb E[X|Y]]=\mathbb E[X]$ and $E[XE[Z|Y]|Y]=E[Z|Y]E[X|Y]$