Modulo Addition is Well-Defined/Proof 1

Proof
We need to show that if:


 * $\eqclass {x'} m = \eqclass x m$
 * $\eqclass {y'} m = \eqclass y m$

then:
 * $\eqclass {x' + y'} m = \eqclass {x + y} m$

Since:
 * $\eqclass {x'} m = \eqclass x m$

and:
 * $\eqclass {y'} m = \eqclass y m$

it follows from the definition of set of integers modulo $m$ that:
 * $x \equiv x' \pmod m$

and:
 * $y \equiv y' \pmod m$

By definition, we have:


 * $x \equiv x' \pmod m \implies \exists k_1 \in \Z: x = x' + k_1 m$
 * $y \equiv y' \pmod m \implies \exists k_2 \in \Z: y = y' + k_2 m$

which gives us:
 * $x + y = x' + k_1 m + y' + k_2 m = x' + y' + \paren {k_1 + k_2} m$

As $k_1 + k_2$ is an integer, it follows that, by definition:
 * $x + y \equiv \paren {x' + y'} \pmod m$

Therefore, by the definition of integers modulo $m$:
 * $\eqclass {x' + y'} m = \eqclass {x + y} m$