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.

Theorem 1 (first page) says that
$$\int_a^b F'(t) dt = F(b) - F(a)$$
so we have
$$\begin{array}{c}G(x) = \int_a^x F'(t) dt = F(x) - F(a) \\
G(x) - F(x) = -F(a)\end{array}$$
But $a$ is constant here, so $F(a)$ is constant. Let $C=-F(a)$ to get the result.
