Riemann-Stieltjes integral with respect to a product of functions.

Question

Let $f, g_1, g_2:[a,b] \rightarrow \mathbb{R}$ be continuous and of bounded variation functions on $[a,b]$. Show that

$$\displaystyle \int_a^b fd(g_1g_2) = \displaystyle \int_a^b fg_1dg_2 + \displaystyle \int_a^b fg_2dg_1$$