Congruence Modulo Real Number is Equivalence Relation

Theorem
Congruence modulo $z$ is an equivalence relation.

Proof
Checking in turn each of the critera for equivalence:

Reflexive
We have that:
 * $$\forall x \in \R: \left({x - x}\right) = 0.z$$

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

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.