When do two functions differ by a constant throughout an interval (Fundamental Theorem of Calculus) I'm reading the proof of the Fundamental Theorem of Calculus here and I don't understand the following parts (at the bottom of page 2):

I don't know how to conclude that $G(x)-F(x)=C$ for a $x \in [a,b]$.
How do I prove the above statement and does it rely on another theorem not mentioned in this proof?
I tried to figure this out by looking at the definitions of $G(x)$ and $F(x)% but only the definition of $G(x)$ is provided.

This is a consequence of the following general fact:

If $f'(x) = 0$ for all $x$ in an interval $[a, b]$, then $f$ is constant on $[a, b]$.

One way to prove this is by the Mean Value Theorem: If there were to exist $x_1$ and $x_2$ in the interval for which $f(x_1) \ne f(x_2)$, there would exist a $c$ between $x_1$ and $x_2$ for which
$$0 \ne \frac{f(x_1) - f(x_2)}{x_1 - x_2} = f'(c)$$
contradicting the fact that $f' \equiv 0$.
