Question on the proof step in the theorem 1 of the Gap statistic paper

From the Gap statistic paper , during the proof for the theorem 1, we can see the below equality (p. 422), $\begin{aligned} \operatorname{var}(X) & =\frac{1}{2} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}(y-x)^2 f(x) f(y) \mathrm{d} x \mathrm{~d} y \\ & =\frac{1}{2} \int_0^1 \int_0^1\left\{F^{-1}(v)-F^{-1}(u)\right\}^2 \mathrm{~d} u \mathrm{~d} v \\ & =\int_0^1 \int_u^1\left\{\int_u^v \frac{1}{f\left\{F^{-1}(t)\right\}} \mathrm{d} t\right\}^2 \mathrm{~d} u \mathrm{~d} v\end{aligned}$ where $F$ and $f$ are unform CDF and PDF, respectively. As I struggled, lower bound in the 3rd line $u$ shuold be $ u= \frac{1}{2}$ to prove the equality between 2nd and 3rd lines. Do I think wrong? Thank you for any idea.