Firstly, I have the same question about this.
The last line of the proof is completely proved if we show that (i) $||f||_2=||f^\vee||_2$ for all $f\in L^2$ and (ii) $f=(f^\wedge)^\vee $, neither of which is to be understood from the answer in the above link.
Above all, the second half of Ch 9 of the textbook is devoted to preparing for the ingredients of this theorem, and all ingredients are about the usual Fourier transform, not the inverse Fourier transform. Thus, I cannot understand how the answer in the link can so easily generalize the statements of the theorem to the case of the inverse Fourier transform. I can hardly believe. Should we prove the last line only in this way? Or, is there another way that is much more understandable? Thanks.