The underlined portion includes part of a list, and each of the three items in the must be in parallel form. The sentence as written uses "proving ..., description of ..., and the furthering of ...." This list is not in parallel structure since the first item ("proving") is a gerund without an article, while the second ("description") is a noun, and the third ("the furthering") is a gerund with an article. Eliminate (A).
With parallel list errors, it's often not possible to scan vertically and group the answers, so inspect each one.
(B) uses a gerund ("proving") without an article, a noun ("description"), and another gerund ("furthering") without an article, so it is not parallel.
(C) correctly uses parallel form by using gerunds without articles for all three items: "proving ..., describing ..., and furthering." It appears to be correct, but check the others.
(D) uses two gerunds ("completing" and "describing") without an article and a gerund ("the furthering of") with an article. So (D) is not parallel.
(E) uses a noun ("the proof") and gerunds ("describing" and "furthering") without articles. Therefore, (E) is not parallel.
(C) is indeed correct.