Mellin Transform of Dirac Delta Function by Function

Theorem
Let $f:\R \to \R$ be a function.

Let $\delta_c\left({t}\right)$ be the Dirac delta function.

Let $c$ be a positive constant real number.

Let $\mathcal M$ be the Mellin transform.

Then:
 * $\mathcal M \left\{ {\delta_c\left({t}\right)f\left({t}\right)} \right\} \left({s}\right) = c^{s-1} f\left({c}\right)$