I am studying over Theorem 11.1.6 of High-Dimensional Probability, by Roman Vershynin.
The theorem states that $f(Ax) - \mathbb{E} f(Ax)$ with $f$ positive-homogeneous and subadditive and $A$ being $m \times n$ Gaussian random matrix with i.i.d. $N(0, 1)$ entries has sub-gaussian increments. In other words, the theorem states that $$\|X_x - X_y\|_{\psi_2} \leq Cb \|x - y\|_2$$ for all $x, y \in \mathbb{R}^n$.
In the proof Vershynin only focuses on the case where $\|x\|_2 = \|y\|_2 = 1$.
I understand all but one part of his proof.
My question: He proved that $\|f\|_{Lip} \leq \|v\|_2$ in the second part of the proof. In his proof asserted that for fixed $t, s \in \mathbb{R}^m$, $$f(t) - f(s) = f(a + \|v\|_2 t) - f(a + \|v\|_2 s).$$
I don't really see why this equality holds. Here $a = Au$, where $u := \frac{x + y}{2}$ and $v := \frac{x - y}{2}$.
Edit: Based on the proof by [Sch06] Gideon Schechtman. Two observations regarding embedding subsets of euclidean spaces in normed spaces. I'm suspecting the $f$ on the LHS (as a function of g) and the $f$ on the RHS (the original $f$ in the statement) are two different functions?
For reference, here's are the screenshots of the theorem and its proof (question is highlighted in blue):
I believe the reasoning goes as follows.
First notice that, since $u$ and $v$ are orthogonal, $\Vert x \Vert_2= \Vert u+v \Vert_2 = \Vert u \Vert_2 + \Vert v \Vert_2 = 1$. Then let $b := Av$ and again for rotational invariance we have $Au + b = b + \Vert u \Vert_2 g \Rightarrow Au = \Vert u \Vert_2 g$. Hence if, for instance, $t = g$ we can write
$$ f(t) \overset{\text{factor 1}}{=} f\big(t \; (\Vert u \Vert_2 + \Vert v \Vert_2)\big) = f\big(t \;\Vert u \Vert_2 + t \; \Vert v \Vert_2\big) = f\big(Au + t \; \Vert v \Vert_2\big) = f\big(a + t \; \Vert v \Vert_2 \big) $$