This is an easy consequence of Euler-Fermat theorem:

Let $d_i,\ m_i$ be defined as in the parent entry. Then $\gcd(a,m_s)=1$ and Euler's theorem implies:

$$a^{\phi(m_s)}\equiv 1 \mod m_s$$

Note also that each of $d_{s-1},\ldots, d_0$ divides $a$ , so $\prod d_i$ divides $a^s$ , so $\prod d_i$ divides $a^{\phi(m_s)+s}-a^s$ . Also, $\gcd(\prod d_i,m_s)=1$ and $m_s\cdot\prod d_i =m$ . Therefore:

$$a^{\phi(m_s)+s}\equiv a^s \mod m$$

which is what the corollary claimed.