In this chapter, we elaborate the art of formal reasoning as the basis of precise argumentation.We start by discussing the general concept of a “proof” as an irrefutable argument why a particular statement is true. We then discuss the formalization of this concept by “proof trees”, i.e., by derivations in a system of inference rules. The focus of this chapter is on a proof system for first-order logic, which is presented both in a formal style (as is required for computer-supported proving) and in an informal style (as is typical for mathematical texts). Finally we summarize various forms of the proof principle of “induction”, which is of fundamental importance in computer science where inductive structures play a major role.