Once a theorem is proved, other results often fall out almost immediately. A corollary is a statement that follows directly from a theorem, usually with little extra work. It is not a new theorem so much as a natural consequence.

Euclid's Elements is full of corollaries. After proving that the angles of a triangle sum to 180 degrees, a corollary states that a triangle cannot have two right angles. The corollary is useful and easy to see once the main theorem is established.

Mathematicians use corollaries to organize knowledge. They highlight important consequences and make them easy to reference. The distinction between theorem, proposition, lemma, and corollary is partly one of emphasis. A lemma is a stepping stone; a corollary is a payoff. Both depend on the logical structure built by the main proof.

• Follows readily from a theorem
• Usually requires little additional proof
• Common in Euclid and modern texts
• Highlights important consequences