proving the fundamental theorem of arithmetic

Question

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 ?

Answer 1

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 $n>pp'$.