Let $\displaystyle \alpha$ be an increasing function on $\displaystyle [a,b], \quad f,g \in \mathcal{R}[\alpha;[a,b]]$ and g is non negative on [a,b]. Suppose $\displaystyle m= \inf \{f(x): a \leq x \leq b \}$ and $\displaystyle M= \sup \{f(x) : a \leq x \leq b\}$. Show that there is a $\displaystyle \lambda \in [m,M] $ such that $\displaystyle \int\limits_{a}^{b} fg \ d\alpha = \lambda \int\limits_{a}^{b} g \ d\alpha$