The book is trying to prove the fundamental theorem of arithmetic but does he mean by saying that at least one of the inequality must be true with strict inequality ? And from where did he come up with this last inequality ?
Since $p\ne p'$, you cannot have both $n=p^2$ and $n=p'^2$. Thus, say, $n>p'^2$.
If we multiply the inequalities, we get $n^2>p^2p'^2$, so $n>pp'$.