I am trying to self study measure theory using Royden's Real Analysis. In Section 6.2, they define the divided difference function $\text{Diff}_h f$ and average value function $\text{Av}_h$ of $[a, b]$ by $$ \text{Diff}_h f(x) = \frac{f(x+h) - f(x)}{h}$$ and $$\text{Av}_h f(x) = \frac{1}{h} \int_x^{x+h} f \text{ for all } x \in [a, b].$$
The text then says, "by change of variables in the integral and cancellation", for all $a \leq u < v \leq b$, $$ \int_u^v \text{Diff}_h f = \text{Av}_h f(v) - \text{Av}_h f(u).$$
I am having some trouble filling in the details of this claim. I am trying to do substitutions of $u = x$ and $v = x+h$, but I'm not getting that the two sides are equal. And I am not seeing the need to do any "cancellation" so I definitely suspect I am doing things incorrectly. Are these the correct changes of variables? Can someone give me a tip about how to fill in the details here?