So far we have called each of the statements to be proved simply an example; in practice, mathematicians use different terms to distinguish different ‘‘levels’’ of result. The conventional terms are:
a theorem is an important result that has been
proved;
a proposition is a less important result that has been
proved;
a lemma is a helpful ‘‘stepping-stone’’ which is proved
in the process of establishing a more important result;
a corollary is an easy consequence of another result
that has been proved;
a conjecture is something that the author thinks is
likely to be true, but has not yet been proved;
a counterexample is an example that shows that a conjecture or similar statement is false.
Another key term in the mathematician’s vocabulary is definition; this is a formal statement of what exactly we mean by a certain word or symbol.