1. **State the problem:** Prove that the difference of any two even integers is even. 2. **Recall the definition of even integers:** An integer $n$ is even if it can be written as $n = 2k$ where $k$ is an integer. 3. **Let the two even integers be:** $a = 2m$ and $b = 2n$, where $m$ and $n$ are integers. 4. **Find the difference:** $$a - b = 2m - 2n$$ 5. **Factor out the common factor 2:** $$a - b = 2(m - n)$$ 6. **Since $m - n$ is an integer (difference of integers is an integer),** let $p = m - n$, where $p$ is an integer. 7. **Rewrite the difference:** $$a - b = 2p$$ 8. **Conclusion:** Since $a - b$ can be expressed as $2$ times an integer $p$, the difference $a - b$ is even. Thus, the difference of any two even integers is even.