Let’s prove the statement “The square of any odd integer is also odd.”
First we restate the theorem in a way that will lead us towards the proof, by introducing a variable into the statement of the theorem to represent “any (given) odd integer”:
Theorem: If is odd, then is also odd.
In order to prove this, we will need the definition of what it means for a integer to be odd:
Definition: An integer is odd if for some integer .
Proof of theorem: Assume is an odd integer. Therefore, by definition, for some . Then
This shows that is odd, since , where , so satisfies the definition of being odd.