I'm looking for a proof of the following classical result by de la Valleé-Poussin.
Let $I=[a,b],\,J=[c,d]$ be two compact intervals. Assume that $u:I\mapsto J$ and $f:J\mapsto\mathbb{R}$ are absolutely continuous. Then, the composition $f\circ u:I\mapsto\mathbb{R}$ is absolutely continuous if and only if $f'\circ u \cdot u'$ is Lebesgue integrable.
The only reference I could find was the original paper of de la Valleé-Poussin from 1915, but it is in French and I can't read it. Are there any 'modern' proofs of the theorem, preferably in English?
Also, I find the expression $f'\circ u \cdot u'$ quite puzzling: If $y_0=u(x_0)$ and $f$ is not differentiable at $y_0$, the expression $f'(u(x_0))=f'(y_0)$ has no meaning. How should we interpret the expression $f'\circ u \cdot u'$ in the 'if part' of the theorem?