Result
R1073: Mean value theorem shows that there exists $c \in (a, b)$ such that
\begin{equation}
f(b) - f(a) = f'(c)(b - a)
\end{equation}
Applying result
R1108: Monotonicity of supremum, we then have
\begin{equation}
|f(b) - f(a)| = |f'(c)||b - a| \leq |b - a| \sup_{x \in (a, b)} |f'(x)|
\end{equation}
This concludes the first claim. Suppose then that $f'$ is bounded with a lower bound $m \in \mathbb{R}$ and an upper bound $M \in \mathbb{R}$. Then results
imply
\begin{equation}
f(b) - f(a) = f'(c) (b - a) \leq M (b - a)
\end{equation}
as well as
\begin{equation}
f(b) - f(a) = f'(c) (b - a) \geq m (b - a)
\end{equation}
$\square$
