Suppose: a = b

It then follows that
a * a = a * b
a^2 = a * b
a^2 - b^2 = a * b - b^2
(a - b) * (a + b) = (a - b) * b
a + b = b
a + a = a
2 * a = 1 * a
2 = 1

Is this derivation really correct? Or is there somewhere a mistake? If so, where?

A woman lived in an apartment building on the seventh floor. Every morning she took the lift to the ground floor and went to her work. But when she came home, she took the lift to the fourth floor and then continued by walking the stairs up. Why would she do that?

A sultan has 14 daughters. He decides to tell every night four of his daughtes a fairy tale, but in such a way that every night, there will be another combination of four daughters. How many nights will keep the sultan busy telling fairy tales?