Cauchy-Riemann Equations - why $f'(z_o) = \frac{\partial f}{\partial x}(z_o)$ implies that f is differentiable at $z_o$

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I'm trying to understand part b of this proof. The only line I don't understand is the sentence starting with "To prove the statement in (b)..."

If someone could clarify why that line is true I could ask a good follow up question.