I was reading Munkres's Analysis of Manifolds. I got stuck on a lemma. I am stating this lemma here.
Let $A$ be an open set in $R^n$ and let $f:A \to \Bbb{R}^n$ be of class $C^1$. If $Df(a)$ is non-singular , then there exist a $m>0$ such that $|f(x_2) - f(x_1)| \geq m |x_2 - x_1|$ holds for all $x_1 , x_2 \in C(a, d)$ (an open ball centered at $a$).
Proof: Let $E= Df(a)$. Then $E$ is non-singular. We first consider the linear transformation that maps $x$ to $Ex$. We compute $$|x_2 - x_1| = |E^{-1}(Ex_2 - E x_1)| \leq n \cdot |E^{-1}||(Ex_2 - E x_1)|.$$
I cannot understand what $|E|$ is here and how this: $$|x_2 - x_1| = |E^{-1}(Ex_2 - E x_1)| \leq n \cdot |E^{-1}||(Ex_2 - E x_1)|$$ happens?