proof of a corollary to Euler-Fermat theorem

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:

$$a^{\varphi (m_s)}\equiv 1mod{m}_s$$

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

$$a^{\varphi (m_s)+s}\equiv a^{s}modm$$

which is what the corollary claimed.