Congruent Numbers are not necessarily Equal

Theorem
Let $x, y, z \in \R$ be real numbers such that:
 * $x \equiv y \pmod z$

where $x \equiv y \pmod z$ denotes congruence modulo $z$.

Then it is not necessarily the case that $x = y$.

Proof
Proof by Counterexample:

We have that:


 * $11 - 5 = 6 = 3 \times 2$

and so by definition of congruence modulo $2$:
 * $10 \equiv 4 \pmod 2$

But $11 \ne 5$.

Also see

 * Equal Numbers are Congruent