How can 1 = 0 ?

Logic Level 2

Peter has enough! All these mathematics do not make any sense to him and he is ready to prove to his teacher that mathematics is based on illogical principles! The way he intends to do this, is by proving that 1=0. Here is his argument :

{1} Suppose that $$x$$ satisfies : $$x-1 = 0$$.
{2} But then that $$\implies (x-1)(x-2) = 0$$
{3} which implies $$x-1=0 \quad \text{ or } \quad x-2 = 0$$
{4} which implies $$x=1 \quad \text{ or } \quad x=2$$
{5} So, in particular, $$x=2$$ is a solution to the first equation and hence, by substituting in, we obtain : $1=0$

Can you tell poor Peter where his argument goes wrong?

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