Modulo Addition is Linear

Theorem
Let $m \in \Z_{> 0}$.

Let $x_1, x_2, y_1, y_2, c_1, c_2 \in \Z$.

Let:
 * $x_1 \equiv y_1 \pmod m$
 * $x_2 \equiv y_2 \pmod m$

Then:
 * $c_1 x_1 + c_2 x_2 \equiv c_1 y_1 + c_2 y_2 \pmod m$

Proof
By Scaling preserves Modulo Addition:
 * $c_1 x_1 \equiv c_1 y_1 \pmod m$
 * $c_2 x_2 \equiv c_2 y_2 \pmod m$

and so by Modulo Addition is Well-Defined:
 * $c_1 x_1 + c_2 x_2 \equiv c_1 y_1 + c_2 y_2 \pmod m$