This paper provides an account of how mathematical knowledge is represented, reasoned about, and used computationally in a mechanized constructive theorem proving environment. We accomplish this by presenting a particular theory developed in the Nuprl proof development system: finite set theory culminating in Ramsey's theorem. We believe that this development is interesting as a case study in the relationship between constructive mathematics and computer science. Moreover, the aspects we emphasize-the high-level development of definitions and lemmas, the use of tactics to automate reasoning, and the use of type theory as a programming logic-are not restricted in relevance to this particular theory, and indicate the promise of our approach for other branches of constructive mathematics.