Okay. It doesn't seem that anyone else is going to add anything. So, since the weekend is almost over and just for closure's sake I'll explain.

Oleg pointed out that in line 3) of the "proof" we were multiplying both sides by zero. This line was arrived at by factoring the (a - b) term out of both sides of the equation. And at this point the proof is still valid because both sides of the equation are indeed still equal (0 = 0).

The problem comes when we take the next step and declare (a + b) = b. It is common practice to eliminate common terms on both sides of an equation, so common in fact that most people never stop to think about just what it is they're doing when they do so. When eliminating common terms, we are actually dividing by that common term on both sides. Normally, anything divided by itself is 1. But, in this case the common term is (a - b) which equals zero; in this case we were dividing by zero. This "proof" is a perfect example of why division by zero is undefined.

That's the rule that was broken; dividing, not multiplying, by zero.