Prove equivalence of two regex using basic identities.

Question

I'm trying to prove the following identity

$$ (x+y)^* = (x^*y)^*x^* = x^*(yx^*)^* $$

Using the following 12 identities

1. $L + M = M + L$
2. $(L + M) + N = L + (M + N)$
3. $(LM)N = L(MN)$
4. $\emptyset + L = L + \emptyset = L$
5. $\epsilon L = L \epsilon = L$
6. $\emptyset L = L \emptyset = \emptyset$
7. $L(M + N) = LM + LN$
8. $(M+N)L = ML + NL$
9. $L + L = L$
10. $(L^*)^* = L^*$
11. $\emptyset^* = \epsilon$
12. $\epsilon^* = \epsilon$

My problem is that I cannot seem to get rid of the $*$ on the right side, no matter what I do. Any hint would really be appreciated.

Answer 1

No, you cannot obtain your identity this way. Indeed, suppose you define $L^*$ by $L^* = \varepsilon + L$. Then the 12 identities are satisfied, but the new wanted one fails to be true.