Cauchy Riemann conditions/equation

[MartinV05]

In the proof of the the Cauchy-Riemann's conditions we have and equality between differentials of the same function (f(z)) by x(real part) and by iy(imaginary part?).
Why do we "say" that both differentials should be equal when it's normally possible to have different differentials according to the variable used?

Picture related (the equality in the last part):

 

[HallsofIvy]

There is only one variable here- z. And the derivative is with respect to that variable. "x" and "y" are not independent variables, they are the real and imaginary parts of the single variable z.

Here they are just taking the limit in "\lim_{h\to 0}(f(z+h)- f(z))/h" as h approaches 0 in different ways. If the limit itself exists, then the limit as h approaches 0 in any way must be the same. You can do the same in differentiation of functions of a real variable- the limits as h goes to 0 "from above" and "from below" must be the same. It is the fact that the complex plane is two dimensional while the real line is only one dimensional the gives more restrictions on the complex derivative.

 

[MartinV05]

So no matter how z changes (whether real or imaginary) the limit should be the same, because they both actually complete the other "higher" variable. I know the that approaching from 0- and 0+ should be the same with limits in real variables, but never would have thought of this in that way. Think you!