Both Opposite Statements Are True. Can YOU spot the mistake?

x=y and NONE are equal to 0

multiply "x" to both sides and you get "x^2 = xy"

subtract y^2 from both sides and you get "x^2 - y^2 = xy - y^2"

Factorize and you get "(x+y)(x-y)=y(x-y)"

Divide both sides by x-y and you get "x+y=y"

Finally, subtract y from both sides and you get "x=0??"

Question marks are there after x=0 because on the top of the paste I said x does not equal 0 and at the final step we said x DOES equal 0. They are two opposite statements which both CANNOT be true.