Proof that congruence is compatible with addition

Proof

1. Let $n$ be a positive integer and let $a$, $b$, $c$, $d$ be integers with $a \equiv b \pmod n$ and $c \equiv d \pmod n$.
2. By the definition of congruence modulo n, $n \mid a - b$ and $n \mid c - d$.
3. By the definition of divisibility, there are integers $k$ and $l$ with $a - b = n k$ and $c - d = n l$.
4. Then $(a + c) - (b + d) = (a - b) + (c - d) = n k + n l = n (k + l)$.
5. Since $k + l$ is an integer, $n \mid (a + c) - (b + d)$.
6. Hence $a + c \equiv b + d \pmod n$.