Why do you emphasize writing here?

I emphasize writing in this course for several reasons.

Firstly, proof has a social component to it. Proving something alone is not enough to be useful. Experts must disseminate that information in ways that other experts, as well as the intended audience, can make use. Much of the benefit of certified-correct software is that we know in such critical applications, that it is correct.

Secondly, writing is critical to capably and effectively communicating your ideas. Even, and in my experience especially in a STEM field, is precision in language important. Exophoric references and an overabundance of pronouns can frustrate your reader and perhaps lead him to make a catastrophic error.

Thirdly, clear writing is highly correlated with clear thinking. The sense of frustration, when writing, at being forced to state explicitly and precisely what had been perfectly sufficient gauzy or muddled thoughts is a relative of the Rubber Duck debugging method.

For more on the importance of writing clearly even in STEM fields, I suggest this post from the Chronicle of Higher Education.