A very simple example

Let us start with a definition

A number $n$ is called even if $n\%2 = 0$. An odd number $m$ is odd if $m \%2 \neq 0$. Futhermore lets write an $m$ as $m=2q+1$ and $n$ = $n = 2p$ for $p,q\in \mathbb{R}$

In order to prove the next theorem we will need the next Lemma which we will not prove.

An even number plus an even number results in an even number.

We can now come to our theorem.

The sum of an odd number $m$ and an even number $n$ is always odd.

Lets write $m = 2q+1$ and $n = 2p$. $\implies$ $m+n$ = $2q+2p + 1 $ = $2(p+q) + 1$. We know $2(p+q) %2 = 0$ but $1\%2 \neq0$ $ \implies$ $m+n$ must be odd.