I was invited to give a talk at a celebration of the 80th birthday of Richard Palais. It was at a celebration of his 60th birthday that I first gave a talk about how to write a proof–a talk that led to [101]. So, I thought it would be fun to give the same talk, updated to reflect my 20 years of experience writing structured proofs. The talk was received much more calmly than my earlier one, and the mathematicians were open to considering that I might have something interesting to say about writing proofs. Perhaps in the last 20 years I have learned to be more persuasive, or perhaps the mathematicians in the audience had just grown older and calmer. In any case, they were still not ready to try changing how they write their own proofs.

My experience preparing and giving the talk made me realize it was time for a new paper on the subject. This paper is built around a simple example–a lemma from Michael Spivak’s calculus text. I tried to show how a mathematician can easily transform the proofs she now writes into structured proofs. The paper also briefly describes how formal structured proofs are written in TLA+, and an appendix contains a machine-checked proof of Spivak’s lemma. While mathematicians will not write formal proofs in the forseeable future, I argue that learning how to write them is a good way to learn how to write rigorous informal proofs.