Show that the statement of conditional independence
$${\textbf{P}}(X,Y | Z) = {\textbf{P}}(X | Z) {\textbf{P}}(Y | Z)$$
is equivalent to each of the statements
$${\textbf{P}}(X | Y,Z) = {\textbf{P}}(X | Z) \quad\mbox{and}\quad {\textbf{P}}(Y | X,Z) = {\textbf{P}}(Y | Z)\ .$$
Show that the statement of conditional independence $${\textbf{P}}(X,Y | Z) = {\textbf{P}}(X | Z) {\textbf{P}}(Y | Z)$$ is equivalent to each of the statements $${\textbf{P}}(X | Y,Z) = {\textbf{P}}(X | Z) \quad\mbox{and}\quad {\textbf{P}}(Y | X,Z) = {\textbf{P}}(Y | Z)\ .$$