Equivalent regular expressions - Proof
Using $s^* = \epsilon + s^*s$, we have
$$
0^+(10^+0)^*0^* = \\
0^+ (\epsilon + (10^+0)^*10^+0)0^* = \\
0^+0^* + 0^+(10^+0)^*10^+00^*
$$
Similarly,
$$
0^+(10^+0)^* =
0^+ (\epsilon + (10^+0)^*10^+0) = \\
0^+ + 0^+(10^+0)10^+0
$$
Therefore it suffices to convince oneself that $0^+0^* = 0^+$ and $0^+00^* = 0^+0$.
If you want to turn it into a completely formal proof, you'll need to satisfy a list of axioms and derivation rules, but the proof will likely follow similar steps.