Microeconomics theory and integration by parts (proof).
Integrate the $f(t)$ (a primitive of which is $F(t)$) and differentiate the $t$. This yields

\begin{align*} 
x(z) F(z) &amp; = \int_{0}^{z}{t f(t)dt} \\
&amp; = \big[ t F(t) \big ]_{t=0}^{t=z}-\int_{0}^{z}{F(t)dt} \\
&amp; = z F(z)-\int_{0}^{z}{F(t)dt}.
\end{align*}

Dividing by $F(z)$ on both sides yields the result.