# optimism error rate calculation

As posted here in the book Elements of Statistical Learning we drive to the following: $$w = \frac{2}{N}\sum_1^Ncov(y_i, \hat{y_i})$$

The derivation is based on the following properties

1. $$E_y E_{Y^0}[(Y_i^0)^2] = E_y[y_i^2]$$
2. $$2 E_y E_{Y^0} [Y_i^0 \hat{y}_i]=2 E_y [E_{Y^0} [Y_i^0]E_{Y^0}[\hat{y}_i]]=2 E_y [y_i] E_y[ \hat{y}_i]$$

My question is how exactly do we prove these properties? As far as I've figured out for the first statement we use tower property of conditional expectation. For the second one we use independence of $$Y_i^0$$ and $$\hat{y_i}$$.

Is this fair or shall I look into some other explanation?