Chapter 5.4, Problem 189E

Let F. G and H be continuous functions on $\left[a,b\right]$

$\left[c,d\right]$ and $\left[e,f\right]$ , respectively, where $a,b,c,d,e$ and $f$ are real number such that $a<b,c<d$ , and $e<f$ .Show that

   ${\displaystyle \underset{a}{\overset{b}{\int }}{\displaystyle \underset{c}{\overset{d}{\int }}{\displaystyle \underset{e}{\overset{f}{\int }}F\left(x\right)G\left(y\right)H\left(z\right)dzdydx}=\left({\displaystyle \underset{a}{\overset{b}{\int }}F\left(x\right)dx}\right)\left({\displaystyle \underset{c}{\overset{d}{\int }}G\left(y\right)dy}\right)\left({\displaystyle \underset{e}{\overset{f}{\int }}H\left(z\right)dz}\right).}}$