Modulo Multiplication is Well-Defined/Warning

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

Let:
 * $a \equiv b \pmod z$

and:
 * $x \equiv y \pmod z$

where $a, b, x, y \in \R$.

Then it does not necessarily hold that:
 * $a x \equiv b y \pmod z$

Proof
But it is not necessarily the case that:
 * $b k_2 + y k_1 + k_1 k_2 z$

is an integer.

In fact, $b k_2 + y k_1 + k_1 k_2 z$ can only be guaranteed to be an integer if each of $b, y, z \in \Z$.

Hence $a b$ is not necessarily congruent to $x y$.