Prove that ordinal multiplication $\alpha \cdot (\beta \cdot \gamma) = (\alpha \cdot \beta) \cdot \gamma$ is associative by using the following facts:
$$\beta \cdot 0=0$$ $$\beta \cdot (\alpha +1)=\beta \cdot \alpha+\beta$$ $$\beta \cdot \alpha=\sup\{ \beta \cdot \gamma \mid \gamma < \alpha \}$$
The second fact is for all $\alpha$ while the third fact is for all limit $\alpha \neq 0$.
I use transfinite induction to prove it. I manage to prove the base case. I stuck at successor case.
My attempt: LHS$\,=\alpha\cdot (\beta \cdot (\gamma+1))=\alpha(\beta \cdot \gamma+\beta)$
RHS$\,=(\alpha\cdot \beta)\cdot(\gamma+1)=(\alpha\cdot \beta) \cdot \gamma + (\alpha\cdot \beta)=\alpha \cdot (\beta \cdot \gamma)+(\alpha\cdot \beta)$
I don't know the distributive law, so I cannot equate LHS to RHS. Can anyone help me?