Corollary

A corollary is a proposition that follows directly from a proven theorem or a general principle, often with little or no additional proof required. In mathematics and logic, corollaries are statements whose truth rests on earlier results and are typically presented after the theorem they depend on. The term comes from the Latin corollarium, meaning something given along with a reward, and in this context it conveys a result that flows from an earlier one.

A corollary is not the same as a lemma or a proposition, though the distinctions can be nuanced. A lemma is generally an auxiliary result used to help prove a larger theorem, while a corollary is a natural, often immediate consequence of a theorem. A proposition or theorem is usually of broader or more central interest; corollaries are derived conclusions that follow from them.

Examples help illustrate the idea. From Euclid’s theorem that there are infinitely many primes, one can derive corollaries such as: for any finite list of primes there exists a prime not on the list, and there is no largest prime. In calculus, a common corollary of the mean value theorem is that a differentiable function with equal endpoints has a point where the instantaneous rate equals the average rate; another corollary is that a function with derivative zero everywhere on an interval is constant.

In mathematical writing, corollaries are often labeled Corollary 1, Corollary 2, and so on, and they serve to highlight important, readily deduced consequences of previously established results. The concept is also used informally to denote conclusions that follow naturally from established ideas.