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