MATRIX ANALYSIS FOR SCIENTIST AND ENGINEERS by Alan Laub This is for chapter 4, question 4. Can somebody help explain to me this proof?

MATRIX ANALYSIS FOR SCIENTIST AND ENGINEERS by Alan Laub

This is for chapter 4, question 4. Can somebody help explain to me this proof? See image for the question as I am not able to properly type the question out. This chapter is about the Moore-Penrose Pseudoinverse.