Let $S$ be the set of odd integers greater than $1$. For each $x \in S$, denote by $\delta (x)$ the unique integer satisfying the inequality 2^{\delta (x)}$a,b \in S$, define: a \ast b\equal{}2^{\delta (a)\minus{}1} (b\minus{}3)\plus{}a. Prove that if $a,b,c \in S$, then: $(a)$ $a \ast b \in S$ and $(b)$ (a \ast b)\ast c\equal{}a \ast (b \ast c).