poincaré inequality direct proof I don't understand an estimate in my textbook, maybe you can help me out!

Lemma (Poincaré's inequality).
Let $Ω ⊂ (0,L) × \mathbb R^{n−1}$. For $u ∈ C^\infty_c(Ω)$ we have the estimate $$
\int_Ω |u|^2\, dx ≤ L^2 \int_Ω |∇u|^2 \, dx.$$

Proof.
Extend $u$ with $u(x) = 0$ for $x \not \in Ω$. For $x = (x^1,x′) ∈ Ω$ estimate $$ |u(x^1,x')|^2 = \bigg|\int_0^{x_1} \frac{\partial u}{\partial x_1} (s,x') \, ds \bigg|^2 \leq \bigg(\int_0^L \bigg|\frac{\partial u}{\partial x_1}(s,x')\bigg| \, ds \bigg)^2 \leq L \int_0^L |\nabla u(s,x')|^2 \, ds.$$
Then it follows
$$ \int_\Omega |u|^2 \, dx \leq \int_{\mathbb R^{n-1}} \int_0^L |u(x_1,x')|^2 \, dx_1 \, dx' \leq \int_{\mathbb R^{n-1}} L^2 \int_0^L |\nabla u(s,x')|^2 \, ds \, dx' \leq  L^2 \int_\Omega |\nabla u|^2 \, dx.$$
$\Box$
I very well understand the first inequalities, via Hölder's inequality.
But in the second line i don't understand where the second $L$ comes from.
And shouldn't the last inequality be an equality?
 A: In the first inequality, integrate with respect to $x_1$ from $0$ to $L$. Since the right hand side is independent of $x_1$ you end up with $$\int_0^L \lvert u(x_1,x')\rvert^2 dx_1 \le L^2 \int^L_0 \lvert \nabla u(s,x') \rvert^2 ds.$$ This is the inequality you apply to derive the second one.
EDIT: The initial inequality that proved is $$\lvert u(x_1,x') \rvert^2 \le L \int^L_0 \lvert \nabla u(s, x') \rvert^2 ds.$$ In this inequality, the left hand side depends on $x_1$, but the right hand side does not. Thus we can integrate with respect to $x_1$ from $0$ to $L$ and switch the order of integration on the right (I suppose this requires Tonelli's theorem): \begin{equation} \begin{aligned} \int^L_0 \lvert u(x_1,x')\rvert^2 dx_1 \le L \int^L_0 \int^L_0 \lvert \nabla u(s,x') \rvert^2 dsdx_1 &= L \int^L_0 \lvert \nabla u(s,x') \rvert^2 \left( \int^L_0 dx_1 \right) ds \\
&=L^2\int^L_0 \lvert \nabla u(s,x') \rvert^2 ds.
\end{aligned} \end{equation} Then we see 
\begin{align*} \int_\Omega |u|^2 \, dx &= \int_{\mathbb R^{n-1}} \left[\int_0^L |u(x_1,x')|^2 \, dx_1\right] \, dx' \\& \leq \int_{\mathbb R^{n-1}} \left[L^2 \int_0^L |\nabla u(s,x')|^2 \, ds\right] \, dx' \\ &=  L^2 \int_\Omega |\nabla u|^2 \, dx.\end{align*} where the inequality of the two terms in the square braces is exactly the inequality we arrived at above.
