Why do we have $p ( x , y | z ) = p ( x | z ) p ( y | x , z )$

Question

Could anyone derive or explain why the formula $p ( x , y | z ) = p ( x | z ) p ( y | x , z )$ is true?

Answer 1

Take the defintion (for $P(B) \neq 0$): $$P(A|B) := \frac{P(A\cap B)}{P(B)}$$ Just use that for every term and you will see that the equation holds.

Edit
I'll make it clearer and use your notation:
$$p(x,y|z)=\frac{p(x,y,z)}{p(z)}$$ and $$p(y|x,z)=\left(\frac{p(y,x,z)}{p(x,z)}= \right)\frac{p(x,y,z)}{p(x,z)}.$$ And this holds by definition.
There is no computation done here.

Answer 2

To get you started. We use: $\mathsf P(\alpha\mid \beta)\;\mathsf P(\beta)=\mathsf P(\alpha\cap \beta)$

$\begin{align} \mathsf P ( x \cap y \mid z ) & = \frac{\mathsf P((x\cap y)\cap z)}{\mathsf P(z)} & \text{if } \mathsf P(z)\neq 0 \\[1ex] & = \frac{\mathsf P((x\cap z)\cap y)}{\mathsf P(z)} \end{align}$

Can you take it from here?