(related to Proposition: Product of Two Odd Numbers)

- Let the integers \(m, n\) be odd.
- By definition, $2\not\mid n$ and $2\not\mid m$ is a not divisor of $n$ and $m.$
- Therefore $n=2k+1$ and $m=2l+1$ for some integers $k,l.$
- For the product, it follows \(nm=(2k + 1)\cdot (2l + 1)=2(kl + k + l) + 1\), because of the distributivity law for integers.
- Therefore, $2\not \mid nm.$
- Thus, the product $nm$ is odd.∎