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## Main Question or Discussion Point

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):

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):