how x y and z become equal in this solution?

Question

I'm trying to understand an example given in my book but not able to understand it as I am quite weak in mathematics. In the below images I don't get how x, y and z become equal to each other. Please help!

Answer 1

You have to show that at $x=y=z$, the derivative is what it is. That means you need to prove the statement

If $x=y=z$, then $\frac{\partial d^2z}{\partial x\partial y} = ...$

And any time you prove a statement of the type "If $A$, then $B$", you do it by assuming $A$ is true and, from that, proving that $B$ is true.