Since $f$ is continuous, result
R2119: Continuous function on closed interval is Riemann integrable guarantees that $f$ is Riemann integrable. Consider therefore the function
\begin{equation}
F : [a, b] \to \mathbb{R}, \quad
F(x) = \int^x_a f(t) \, d t
\end{equation}
Result
R3183: First fundamental theorem of Riemann integral calculus says that $F$ is differentiable on $(a, b)$ and that $F$ agrees with $f'$ everywhere on $(a, b)$. Given this, result
R3184: Second fundamental theorem of Riemann integral calculus shows that
\begin{equation}
F(b) - F(a)
= \int^b_a f(t) \, d t
\end{equation}
Applying
R1073: Mean value theorem to $F$, we now know that there is $c \in (a, b)$ such that
\begin{equation}
\int^b_a f(t) \, d t
= F(b) - F(a)
= F'(c)(b - a)
= f(c)(b - a)
\end{equation}
This establishes the first claim. The second claim follows simply by dividing each side by the nonzero quantity $b - a$. $\square$
