I'm having trouble proving an equation.

Take $\displaystyle f(x)$ to be a continuous strictly increasing function with $\displaystyle f(0)=0$ and let $\displaystyle {f}^{-1}(x)$ be its inverse. Then:

$\displaystyle \int_{0}^{a}f(x)dx+\int_{0}^{f(a)}{f}^{-1}(y)dy=af(a)$

This should be visually obvious. The graph of $\displaystyle f(x)$ divides the rectangle with vertices $\displaystyle (0,0)$ and $\displaystyle (a,f(a))$ in two.

In the proof, I would like not to assume differentiability (since the equation is obviously true non-smooth functions too)

Thanks.