I would like to extend a previous question on the capitalization of theorem names . This time, it's not about numbered theorems like "Lemma 1.1", but about theorems that carry a name.

Question: Which of the following variants is to be preferred?

- "By Zorn's lemma, we can conclude..."
- "By Zorn's Lemma, we can conclude..."

Additionally, I would like to know how to deal with the corner case where the name of the person after which the theorem is named starts with a lowercase letter

Question: Which of the following variants is to be preferred?

- "The proof of van der Waerden's theorem can be extended to ..."
- "The proof of Van der Waerden's theorem can be extended to ..."
- "The proof of van der Waerden's Theorem can be extended to ..."
- "The proof of Van der Waerden's Theorem can be extended to ..."