Question about proof of the inverse function theorem

Question

I am studying this proof (see picture) of the implicit function theorem in multivariable analysis. I understand the proof, but I do not understand why we can assume without loss of generality that $x_0=y_0=0$and f'(0)=Id. I saw in another topic that they gave the hint $F(x)=f'(x_0)^{-1}(f(x+x_0)-y_0)$, but I don't know how to use this. Could somebody please help me? Thank you!