P3329
If $x \in \mathbb{R}$, then
\begin{equation}
\begin{split}
\mathbb{P}(F^{-1}(U) \leq x)
& = \mathbb{P}(F F^{-1}(U) \leq F(x)) \\
& = \mathbb{P}(U \leq F(x)) \\
& = F_U(F(x)) \\
& = F(x)
\end{split}
\end{equation}
where $F_U$ is a distribution function for $U$. $\square$
