This paper aims to reflect on the concept of “authentic mathematical activities” and attempts to understand mathematicians’ activities based on a comparative analysis of books from the field of mathematics. We included Masahiko Saito’s Sugaku no kiso: shugo, su, iso (The Basics of Mathematics: Sets, Numbers, and Topology) and Kazuo Matsusaka’s Shugo, iso nyumon (Introduction to Sets and Topology) in this study, focusing on the difference between the two authors’ descriptions of the definition of continuous mapping and proofs of the equivalence of conditions. We found that (1) things that the latter work thoroughly proved were boldly omitted in the former work on the grounds that they are “trivial” and (2) some cases have been observed where the latter work has used abundant set theory expressions to clearly prove things, whereas in the former work, these are proven without directly using such expressions and with the adoption of intermediate concepts. In accordance with these differences, we infer that detailing proofs is not writing in universal, ideal, impersonal sentences but rather ones that include human and implicit messages for an assumed target audience. This occurs regardless of whether one is proving an as yet unknown theorem or constructing new proofs regarding known theorems while consulting known proofs (reconstructing proofs) . In this paper, we highlight the authenticity of the reconstruction of known proofs while considering a specific audience, which constitutes a suggestion to mathematics education. This is a new perspective that has not received sufficient attention in previous mathematics education research.