A theorem is a statement that has been proved true. It is not assumed or conjectured; it follows from axioms, definitions, and previously proven results by logical deduction. The proof is what makes a theorem reliable.

Famous theorems include the Pythagorean theorem, the fundamental theorem of calculus, and Fermat's Last Theorem. Some theorems are simple; others take hundreds of pages to prove. The four-color theorem was the first major theorem proved with computer assistance.

Theorems are the main results of mathematics. They are named, numbered, and referenced. A theorem may have corollaries, lemmas, and generalizations. The process of proving theorems is the heart of mathematical research.

• Statement proven true by logic
• Follows from axioms and prior results
• Examples: Pythagorean, Fermat's Last
• Proof is what establishes truth