Convergence proof on Newton's method from Boyd & Vandenberghe's book on convex optimization From Boyd &amp; Vandenberghe's book, page 491:

Applying the Lipschitz condition,we have
$$\begin{aligned}
\left\|\nabla f\left(x^{+}\right)\right\|_{2} &amp;=\left\|\nabla f\left(x+\Delta x_{\mathrm{nt}}\right)-\nabla f(x)-\nabla^{2} f(x) \Delta x_{\mathrm{nt}}\right\|_{2} \\
&amp;=\left\|\int_{0}^{1}\left(\nabla^{2} f\left(x+t \Delta x_{\mathrm{nt}}\right)-\nabla^{2} f(x)\right) \Delta x_{\mathrm{nt}} d t\right\|_{2} \\
&amp; \leq \frac{L}{2}\left\|\Delta x_{\mathrm{nt}}\right\|_{2}^{2} \\
&amp;=\frac{L}{2}\left\|\nabla^{2} f(x)^{-1} \nabla f(x)\right\|_{2}^{2} \\
&amp; \leq \frac{L}{2 m^{2}}\|\nabla f(x)\|_{2}^{2}
\end{aligned}$$
to get the inequality assumption (9.33), which is:

$$\frac{L}{2 m^{2}}\left\|\nabla f\left(x^{(k+1)}\right)\right\|_{2} \leq\left(\frac{L}{2 m^{2}}\left\|\nabla f\left(x^{(k)}\right)\right\|_{2}\right)^{2}$$

I dont understand how he get the first two lines of the derivation? 

My understanding is that $\nabla f\left(x^{+}\right)$ equals $\nabla f\left(x+\Delta x_{\mathrm{nt}}\right)$ already when t=1, so where do the rest of two terms come from?
please anyone help me, really appreciate.
$-\nabla f(x)-\nabla^{2} f(x) \Delta x_{\mathrm{nt}}=0$ since $\Delta x _{nt}$ is the Newton search direction.
$-\nabla f(x)-\nabla^{2} f(x) \Delta x_{\mathrm{nt}}=0$ since $\Delta x _{nt}$ is the Newton search direction.