  The main aim of this paper is to promote a certain style of doing coinductive
proofs, similar to inductive proofs as commonly done by mathematicians. For
this purpose, we provide a reasonably direct justification for coinductive
proofs written in this style, i.e., converting a coinductive proof into a
non-coinductive argument is purely a matter of routine. In this way, we provide
an elementary explanation of how to interpret coinduction in set theory.
