Lebesgue differentiation theorem for monotone functions via Vitali covering lemma

Question

I was reading LECTURES ON LIPSCHITZ ANALYSIS by Juha Heinonen and at the Theorem 3.2 he gives a proof of Lebesgue Differentiation Theorem for monotone functions. He says that we can easily (using the Vitali covering lemma and the definitions of $D^+f(x)$ and $D^-f(x)$) check the following:

If $E_q = \{ x \in (a,b); D^+f(x) > q \}$ and $E_p = \{ x \in (a,b); D^-f(x) < p \}$ then

1. $q|E_q| \leq |f(E_q)|$,
2. $|f(E_p)| \leq p|E_p|$.

But I couldn't prove it on my own. I found the calculations here but I was wondering if has another way more direct for check it "easily" like the book claim.

Thanks!

Notes:

1. f is a increasing functions,
2. $|\cdot|$ is the Lebesgue measure,
3. $D^+f(x) = \limsup_{h \to 0} \frac{f(x+h)-f(x)}{h}$,
4. $D^-f(x) = \liminf_{h \to 0} \frac{f(x+h)-f(x)}{h}$.