# Path from variance of product to product of variance and expectation

In one paper authors tell:

We let the initialised elements in $$W_l$$ be independent and identically distributed (i.i.d.). We assume that the elements in $$x_l$$ are also i.i.d. and $$x_l$$ and $$W_l$$ are independent of each other. Then we have:

$$Var[y_l]=n_lVar[w_lx_l]$$

We let $$w_l$$ have zero mean. Then the variance of the product of independent variables gives us:

$$Var[y_l] = n_lVar[w_l]E[x_l²]$$

Could someone please describe what might be the way from the first equation to the second?

\begin{align}\operatorname{Var}(w_lx_l)&=\mathbb E[w_l^2x_l^2]-\mathbb E[w_lx_l]^2=\mathbb E[w_l^2]\mathbb E[x_l^2]-\mathbb E[w_l]^2\mathbb E[x_l]^2=\mathbb E[w_l^2]\mathbb E[x_l^2]\\&=\operatorname{Var}(w_l)\mathbb E[x_l^2]\end{align}