\begin{eqnarray*}[a[an]]-[an] &=& [(1 + \phi)[(1 + \phi)n]] - [(1 + \phi)n]\\ &=& [[n + \phi n] + \phi[n + \phi n]] - [n + \phi n]\\ &=& [n + \phi n] + [\phi(n + [\phi n])] - [n + \phi n]\\ &=& [\phi n + \phi[\phi n]]\quad(*)\end{eqnarray*}