Four statements equivalent to well-foundedness (well-founded induction, existence of recursively defined functions, uniqueness of recursively defined functions, and absence of descending omega-chains) have been proved in Mizar and the proofs mechanically checked for correctness. It seems not to be widely known that the existence (without the uniqueness assumption) of recursively defined functions implies well-foundedness. In the proof we used regular cardinals, a fairly advanced notion of set theory. The theory of cardinals in Mizar was developed earlier by G. Bancerek. With the current state of the Mizar system, the proofs turned out to be an exercise with only minor additions at the fundamental level. We would like to stress the importance of a systematic development of a mechanized data base for mathematics in the spirit of the QED Project.

We report on how mathematics is done in the Mizar system. Mizar oers a language for writing mathematics and provides software for proof-checking. Mizar is used to build Mizar Mathematical Library (MML). This is a long term project aiming at building a comprehensive library of mathematical knowledge. The language and the checking software evolve and the evolution is driven by the growing MML.