Problems while trying to prove that $(r|s)^* = (r^*.s^*)^*$
In one direction,
$$
L((r^*s^*)^*) \subseteq L(((r|s)^*(r|s)^*)^*) = L(((r|s)^*)^*) = L((r|s)^*).
$$
In the other direction, $L(r) \subseteq L(r^*s^*)$ and $L(s) \subseteq L(r^*s^*)$ imply that $L(r|s) \subseteq L(r^*s^*)$, and so
$$
L((r|s)^*) \subseteq L((r^*s^*)^*).
$$