Does finding real income adjust for inflation?
Indeed deflating nominal income using CPI does get rid of the effect of inflation. I think you're just getting confused by the phrasing.

I assume you are referring to the following sentence on pg. 6:


  We  have  shown  that  the  net  result  of  her  raise  and  inflation  was  an  increase of 7.9% in real income.


Let me rephrase this sentence: we have shown that after raising her salary by 10% and then adjusting for inflation, the net effect is that her real income increased by 7.9%.