Existence of Translation between Each Pair of Points in Euclidean Space

Theorem
Let $\R^n$ denote the real Euclidean space of $n$ dimensions.

Let $\mathbf a = \tuple {a_1, a_2, \ldots, a_n}$ and $\mathbf b = \tuple {b_1, b_2, \ldots, b_n}$ be points in $\R^n$.

There exists an isometry $f: \R^n \to \R^n$ such that $\map f {\mathbf a} = b$.

Proof
Let $\mathbf t = \mathbf a - \mathbf b$.

Then the translation $\tau_\mathbf t$ is such an isometry.

We have that:
 * $\map {\tau_\mathbf t} {\mathbf a} = \mathbf a - \paren {\mathbf a - \mathbf b} = \mathbf b$

The result follows from Translation Mapping is Isometry.