I am reading a book, it explains the Euler's proof of Fermat's little theorem (FLT). There are 3 theorems are presented to prove FLT, I understood the first two (I will skip the proof of each theorem), but I could not understand the proof of third theorem.
Theorem 1. if $p$ is prime then, $\quad(a+1)^p- (a^p + 1) = 0\mod p$
Theorem 2. $(a+1)^p-(a+1) = 0 \mod p \quad$ if $\quad a^p = a\mod p$
Theorem 3. $a^p = a \mod p$
I understood the first two, but then in Theorem 3 the author proves this step by using Theorem 2. But my confusion is that Theorem 3 was assumed in Theorem 2, looks like a circular reasoning. Is this the Euler's original proof for FLT? What is the Euler's original proof?