Simple present

You should use the simple present tense.

An abstract is normally written in something like the "historical present". Brief summaries of longer works are often written like this. For example, you can see it in TV listings:

Hope returns to Salem with Deimos in tow; Rafe gets bad news from Greece; Sonny, Paul, Chad, Gabi, JJ, Lani and Eli find themselves stranded on an island; Ciara says goodbye to Salem.

^{Source: TV Guide's listing for Days of Our Lives, May 26, 2017.}

and in blurbs on the back of books:

In the remote border country of South Utah, a man is about to be whipped by the Mormons in order to pressure Jane Withersteen into marrying against her will. The punishment is halted by the arrival of the hero, Lassiter, a gunman in black leather, who routs the persecutors and then gradually recounts his own history of an endless search for a woman abducted long ago by the Mormons.

^{Source: Amazon's blurb for Riders of the Purple Sage by Zane Grey.}

and in titles of movies that summarize the plot of the movie:

A Star Is Born

as well as (of course) in abstracts of scientific papers:

In this article, we summarise the current and upcoming versions of SBML and our efforts at developing software infrastructure for supporting and broadening its use. We also provide a brief overview of the many SBML-compatible software tools available today.

^{Source: "The Systems Biology Markup Language." M. Hucka, A. Finney et al. Syst. Biol. Vol. 1, No. 1, June 2004.}

The simple present tense in English seldom refers to the present moment (except for a few special verbs such as linking verbs and have). It leads the reader or listener to "step into another time" or rather "step out of time" altogether, as when stating timeless generalities like "Day follows night." A blurb describes the product—not the product at a certain time, but simply the product—so the simple present tense is a natural choice.

As an introductory summary of a story, or an event that extends across an interval of time, like "A Star Is Born" or a "teaser" narration before a TV news story, the simple present indicates the whole interval covered in the full story. It suggests that the larger work begins at the beginning of that extended event and ends with the completion of that event. Extending this manner of interpretation to an abstract, a reader would understand your use of the simple present to mean that the paper consists of the proof you described—since a proof can easily be understood in a story-like way, beginning with the first proposition and ending with the theorem.

Other tenses in an abstract

None of the above should suggest that only the present tense is appropriate in an abstract, though. Rather, the "historical present" narration gives all tenses a different meaning—a "historical" meaning. Here's more of the abstract I excerpted from above:

Biologists are increasingly recognising that computational modelling is crucial for making sense of the vast quantities of complex experimental data that are now being collected. The systems biology field needs agreed-upon information standards if models are to be shared, evaluated and developed cooperatively. Over the last four years, our team has been developing the Systems Biology Markup Language (SBML) in collaboration with an international community of modellers and software developers. SBML has become a de facto standard format for representing formal, quantitative and qualitative models at the level of biochemical reactions and regulatory networks.

All the main verbs (marked in bold) put the paper into its historical context:

• are increasingly recognizing, are now being collected: the continuous present indicates the current state of the field—always changing.

• needs: simple present, but with a verb that in normal contexts would use the simple present for the immediate moment, so—the current state of the field.

• Our team has been developing: this puts the author's work into historical perspective. The present perfect continuous tense indicates that the team's work started some time ago, has continued to the present time, and is on-going.

• has become: this puts SBML into its historic context: the present perfect tense means that the acceptance by biologists of SBML as a standard is already accomplished fact at the time of publication of the paper. The paper is not proposing that SBML become a standard, it's reporting on the work that enabled it to become a standard.

So, the "present moment" in an abstract is the time at which the paper is published. But for verbs that describe the paper itself, or the results it reports, the present moment is the "eternal now" signified by the simple present tense.

In the body of the paper

In the body of the paper, of course, tenses return to their normal meaning. So, in the conclusion of the paper, to summarize the result presented in previous sections of the paper, you would use your present-perfect sentence. In that context, the present perfect refers to the range of time from the start of the paper to the present sentence.

By the way, if it's a mathematical paper, then you should definitely say a proof, not "proof", since mathematicians always see each proof as just one of many ways to prove the same theorem. "Affirmation" sounds like a strange word to use in mathematics, though. You might want to ask a separate question about that.