There are many ways to prove a theorem, including:
- Reductio ad absurdum: If we can show that the assumption that our hypothesis is false leads to a contradiction, it follows that the hypothesis must be true.
- Mathematical induction
In mathematical logic, a derivation is defined as a sequence of statements, each of which is either 1) an assumption, 2) a tautology, or 3) follows from two previous statements by the rule of modus ponens - the idea is that the statements form a tree, with assumptions and tautologies at the leaves. A theorem is any statement which has a derivation. Of course in practice more complicated rules are also used.
Closely related to theorem proving is automated theorem proving.