Euclidean Metric on Real Number Space is Translation Invariant

Theorem
Let $\tau_{\mathbf t}: \R^n \to \R^n$ denote the translation of the real Euclidean space of $n$ dimensions by the vector $\mathbf t = \tuple {t_1, t_2, \ldots, t_n}$.

Let $d_2$ denote the Euclidean metric on $\R^n$.

Then $d_2$ is unchanged by application of $\tau$:


 * $\forall \mathbf x, \mathbf y \in \R^n: \map {d_2} {\map \tau {\mathbf x}, \map \tau {\mathbf y} } = \map {d_2} {\mathbf x, \mathbf y}$

Proof
Let $\mathbf x = \tuple {x_1, x_2, \ldots, x_n}$ and $\mathbf y = \tuple {y_1, y_2, \ldots, y_n}$ be arbitrary points in $\R^n$.

Then: