Prologue - Problem 2

We have to show what is wrong with the following proof. Let $x = y$. Then: \begin{align*} x^2 &= xy && \text{1} \\ x^2 - y^2 &= xy - y^2 && \text{2} \\ (x + y)(x - y) &= y(x - y) && \text{3} \\ x + y &= y && \text{4} \\ 2y &= y && \text{5} \\ 2 &= 1 && \text{6} \end{align*} In line (3) to (4) we are getting rid of $(x - y)$ by the multiplicative inverse, but that change is illegal, because it would mean a division by $0$.