Mathematical theorem

A mathematical theorem is a statement established by a mathematical proof from specified assumptions and previously justified results. Its hypotheses belong to the claim: proving a conclusion for one class of objects does not automatically prove it for a larger class.

For every nonnegative integer $n$, the sum of the first $n$ odd positive integers is $n^2$. A proof by mathematical induction starts with the empty sum at $n=0$; adding the next term gives $n^2+(2n+1)=(n+1{)}^2$. This argument establishes all cases from one implication. Checking many numerical cases can suggest the result, but does not replace the inductive step unless the full stated domain is finite and exhaustively covered.