We fix two functions $f$ and $g$ in $E$. For $x > 0$, we set
$$F ( x ) = - U ( f ) ^ { \prime } ( x ) \mathrm { e } ^ { - x }$$
where $U(f)^\prime(x) = \mathrm { e } ^ { x } \int _ { x } ^ { + \infty } f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. Verify that $F$ is an antiderivative of $x \mapsto f ( x ) \frac { \mathrm { e } ^ { - x } } { x }$ on the interval $\mathbb { R } _ { + } ^ { * }$.