Let $f:[a,b]\to\mathbb{R}$ be a continuous function and $g:[a,b]\to\mathbb{R}$ be an absolutely continuous function. It is known that the absolutely continuous function $g$ is diffentiable a.e. and there holds $$g(y)-g(x)=\int_x^y g'\;\mbox{d}t$$ for every $x,y\in[a,b]$. Can we somehow use these facts to prove that for the Riemann-Stieltjes integral there holds $$\int_a^b f\;\mbox{d}g=\int_a^b fg'\;\mbox{d}x?$$
Thanks for any help.