Congruence Modulo Real Number is Equivalence Relation

Theorem
For all $$z \in \R$$, congruence modulo $z$ is an equivalence relation.

Proof
Checking in turn each of the critera for equivalence:

Reflexive
We have that Equal Numbers are Congruent:
 * $$\forall x, y, z \in \R: x = y \implies x \equiv y \, \bmod \, z$$

so it follows that:
 * $$\forall x \in \R: x \equiv x \, \bmod \, z$$

and so congruence modulo $z$ is reflexive.

Symmetric
$$ $$ $$ $$

So congruence modulo $z$ is symmetric.

Transitive
$$ $$ $$ $$ $$ $$ $$

So congruence modulo $z$ is transitive.

So we are justified in supposing that congruence, as we have defined it, is an equivalence.