We have,
\begin{equation*}
\mathscr{L}[f(t)] =\int\limits_{0}^{\infty}e^{-st}f(t)\,dt
\end{equation*}
or,
\begin{equation*}
\mathscr{L}[f(at)] = \int\limits_{0}^{\infty}e^{-st}f(at)\,dt
\end{equation*}
put \(at=u\) so that \(\,dt=\frac{1}{a}\,du\text{.}\)
\begin{equation*}
\therefore \quad \mathscr{L}[f(at)]=\int\limits_{0}^{\infty}e^{-su/a}f(u)\,du
\end{equation*}
where
\begin{equation*}
p=s/a = \frac{1}{a}F(p) = \frac{1}{a}F(s/a)\text{.}
\end{equation*}
