Axioms For *
\begin{align} 1 + aa^* &\leq a^* \\ 1 + a^*a &\leq a^* \\ b + ax &\leq x \to a^*b \leq x \\ b + xa &\leq x \to ba^* \leq x \\ \end{align}
Elementary Results
\begin{align} a \leq b &\to a + c \leq b + c \\ a \leq b &\to ac \leq bc\, \wedge\, ca \leq cb \\ a \leq b &\to a^* \leq b^* \end{align}
Problem
Prove the following identity in a Kleene algebra using only the axioms and elementary results. $$(a + ab + b)^* = (a + b)^*$$
Solution: \begin{align} (a + b)^* &= (a + ab + b)^* \\ (a + b)^* &\leq (a + ab + b)^* \\ 1 + (a + b)(a + b)^* &\leq 1 + (a + ab + b)(a + ab + b)^* \\ (a + b)(a + b)^* &\leq (a + ab + b)(a + ab + b)^* \\ \end{align}
Quesiton
So for them to be equal the sets should be contained in each other. At which point do I transition to an inequality?
Is it right to say a $*$ cannot be removed since it has no inverse?
Can I distribute into a $*$? Say $(a +b)(a + b)^*$ or do they need to have the same $*$ height?
Some hints to get further would be greatly appreciated.