Modulo Addition is Closed/Real Numbers

Theorem
Let $z \in \R$ be a real number.

Then addition modulo $z$ on the set of all residue classes modulo $z$ is closed:


 * $\forall \left[\!\left[{x}\right]\!\right]_z, \left[\!\left[{y}\right]\!\right]_z \in \R_z: \left[\!\left[{x}\right]\!\right]_z +_z \left[\!\left[{y}\right]\!\right]_z \in \R_z$.

Also see

 * Modulo Addition is Closed/Integers

Proof
From the definition of addition modulo $z$, we have:
 * $\left[\!\left[{x}\right]\!\right]_z +_z \left[\!\left[{y}\right]\!\right]_z = \left[\!\left[{x + y}\right]\!\right]_z$

As $x, y \in R$, we have that $x + y \in \R$ as Real Addition is Closed.

Hence by definition of congruence, $\left[\!\left[{x + y}\right]\!\right]_z \in \R_z$.