...... (simply subtracting the modulus to each term)

...... (Factor out from each term)

Now, notice that all the numbers from 1 to (p-1) are all least residues of p. We have half the numbers since all the even numbers from 1 to (p-1) are gone, leaving us with odd terms (1, 3, 7, ... , p-2 ) and thus factors of . So:

......