OpenAI prompt for System C

Consider the following research-level math question.

[Question]

Carefully consider the question and provide a complete and rigorous proof. Provide either complete proofs or careful citations for all intermediate statements in the proof. Work very hard to prove the question and do not return until a complete proof is achieved.

Make sure to structure the proof with careful lemmas and theorem statements. The output must be a compilable LaTeX document conforming to the standards of rigor and scholarship prevailing in mathematical literature.