Using result
R1831: Real arithmetic expression for binomial coefficient, we have
\begin{equation}
\begin{split}
\frac{n + 1}{m + 1} \binom{n}{m}
& = \frac{n + 1}{m + 1} \frac{n !}{(n - m) ! m !} \\
& = \frac{(n + 1) !}{(n - m) ! (m + 1) !}
= \frac{(n + 1) !}{((n + 1) - (m + 1)) ! (m + 1) !}
= \binom{n + 1}{m + 1}
\end{split}
\end{equation}
$\square$
