Modulo Addition is Closed/Real Numbers

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

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


 * $\forall \eqclass x z, \eqclass y z \in \R_z: \eqclass x z +_z \eqclass y z \in \R_z$.

Also see

 * Modulo Addition is Closed/Integers

Proof
From the definition of addition modulo $z$, we have:
 * $\eqclass x z +_z \eqclass y z = \eqclass {x + y} z$

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

Hence by definition of congruence, $\eqclass {x + y} z \in \R_z$.