Where is the sophism?

This is a puzzle created by the mathematician Augustus De Morgan. He offered this as a "proof" that 2 = 1, which is (as he understood) absurd, and dependent on sophistical reasoning. Can you spot the flaw?

1. Let x = 1.

2. Clearly,then, x² = x.

3. So x² - 1 = x - 1.

Now, divide both sides of THAT equation by x-1.

That 'should' turn it into the following

4. x + 1 = 1.

But since we started by saying that X equals 1, we have now concluded that

5. 2 = 1. QED.

That plainly cannot be.

But where is the sophism? The answer begins with remembering a critical point in number theory: don't divide anything by zero. Ever.

Any chain of inference that involves the division of anything by zero at any point is invalid.

The first proposition above tells us that x = 1. Later we are instructed to divide both sides of a certain equation by x-1. This is an indirect way of telling us to divide it by zero.

TILT.

Accordingly, we never get to steps four or five.

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