There's a catch somewhere!

"I can prove that 1 = 2," my friend declared. Now, math has always been my weak spot, and I wasn't very keen on listening to his rambling. "I will give you a treat if you catch the error," he said. That caught my attention. My stomach is also my weak spot, and it always demands something or the other. "Okay, go ahead," I said. "Even with my poor knowledge of math I am sure I can detect any error if somebody tries to prove that 1 = 2," I said.

"Let x = y," he began. I liked that; it was familiar ground. In class, the teacher always told us something of that kind. "But, remember that x is not equal to zero," he added.

"Alright," I said.

"Let me multiply both sides by x. Any problem with that?" I could see no problem in multiplying both sides by x.

"So, we have x * x = y * x. That's x² = y * x," my friend said. "Is that okay?" he asked. I said it was fine.

"Now, let me subtract y² from both sides. Since I am subtracting both sides by the same number, the equality holds."

I thought so too and told him to go ahead with the subtraction.

"Well, we now have x² - y² = y * x - y²."

I could see nothing wrong in that.

"Now, the left hand side is x² - y² which can be factorised as (x + y) * (x - y), while in the right hand side y is common and can be taken out to give y * (x - y)," he said.

That x² - y² = (x + y) (x - y) had been drilled into my head in school and I saw nothing wrong in that too. And, anyone knows that y * x - y² = y * (x - y).

"So, we have (x + y) * (x - y) = y * (x - y)."

I nodded my head in agreement.

"Now, I can divide both sides by (x - y)," he said.

"Go ahead, do it," I said.

"So, we are left with x + y = y."

I had to agree.

"We started by saying that x = y, so we replace x with y," he said.

I said nothing this time.

"So, we have y + y = y, or 2y = y. Therefore, we get 2 = 1."

My friend grinned at me and wanted to know whether there was any error.

I am absolutely certain he has pulled a fast one on me, but I can't figure out where is the catch. Please help me discover where the error lies.

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