Previously, it was unnecessarily difficult to parse the sentence to
determine that "HTTP response" and "internal state of the server
following the request" are the coherent ideas. Even if length wasn't
an issue, e.g. "... and checking both the A and B are correct", the
sentence still feels a bit fragile without a "that".
Since the second phrase is indeed relatively long, and "internal state
of the server" is a reasonable guess for the second coherent idea, the
"the" helps to reset the reader's expectation about where the next
coherent idea starts, and ends.
Lastly, having "both" in front of the two phrases encourages an
assumption that they're shorter (which is especially problematic for
the second phrase), while having it at the end of the sentence helps
to anchor the end of the second phrase; this is especially true since
the absence of "both" before that point encourages an assumption that
you haven't finished reading yet, given that two things have been
mentioned.
