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A theorem and its proof are typically laid out as follows: Theorem (name of person who proved it and year of discovery, proof or publication). Statement of theorem (sometimes called the proposition). Proof. Description of proof. End mark. The end of the proof may be signalled by the letters Q.E.D. (quod erat demonstrandum) or by one of the tombstone marks "□" or "∎" meaning "End of Proof", introduced by Paul Halmos following their usage in magazine articles. The exact style depends on the author or publication. Many publications provide instructions or macros for typesetting in the house style. It is common for a theorem to be preceded by definitions describing the exact meaning of the terms used in the theorem. It is also common for a theorem to be preceded by a number of propositions or lemmas which are then used in the proof. However, lemmas are sometimes embedded in the proof of a theorem, either with nested proofs, or with their proofs presented after the proof of the theorem. Corollaries to a theorem are either presented between the theorem and the proof, or directly after the proof. Sometimes, corollaries have proofs of their own that explain why they follow from the theorem. |
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