Congruence (Number Theory) is Congruence Relation

Theorem
Congruence modulo $m$ is a congruence relation on $\left({\Z, +}\right)$.

Proof
Suppose $a \equiv b \bmod m$ and $c \equiv d \bmod m$.

Then by the definition of congruence there exists $k, k' \in \Z$ such that:


 * $\left({a - b}\right) = km$


 * $\left({c - d}\right) = k'm$

Hence:


 * $\left({a - b}\right) + \left({c - d}\right) = km + k'm$

Using the properties of the integers:


 * $\left({a + c}\right) - \left({b + d}\right) = m \left({k + k'}\right)$

Hence $\left({a + c}\right) \equiv \left({b + d}\right) \bmod m$ and congruence modulo $m$ is a congruence relation.