I am studying the proof that if $p \equiv 1 \pmod 4$ then $p$ can be written as a sum of squares. That is one implication of Fermat's two square theorem.
I have stumbled upon this text: https://people.bath.ac.uk/masgcs/book1/amplifications/fermat2sq.pdf
More specifically, it states that:
I don't understand specifically why in the aforementioned excerpt there exist $c$, $d$ so that $x_1 = x - cm_0$ and $y_1 = y - dm_0$. How can we prove it?