Suppose that $F,G \in NBV$ and $-\infty <a<b< \infty$, how show that
$\displaystyle\int_{[a,b]} \dfrac{F(x)+F(x-)}{2}dG(x) + \displaystyle\int_{[a,b]} \dfrac{G(x)+G(x-)}{2}dF(x) = F(b)G(b)-F(a-)G(a-)$
where $F(a-)=\lim_{x \to a^-} F(x) $.
This exercise is from Folland Real analysis. The author suggests that he imitate the proof of Theorem 3.36, but I am not able to use the same lines. Can someone give a tip?