Congruence of Product

Theorem
Let $$a, b \in \mathbb{Z}$$ and $$m \in \mathbb{N}$$.

Let $$a$$ be congruent to $b$ modulo $m$, i.e. $$a \equiv b \left({\bmod\, m}\right)$$.

Then $$\forall c \in \mathbb{Z}: c a \equiv c b \left({\bmod\, m}\right)$$.

Proof
From Congruence Modulo m Equivalence, $$c \equiv c \left({\bmod\, m}\right)$$.

Thus from the definition of Multiplication Modulo m, the result follows.