Proposition 4

Let $a$ and $b$ be integers. Then the mapping $x \rightarrow a + x$ is a strictly inrceasing isomorphism of the interval $[0, b]$ onto the interval $[a, a + b]$, and $y \rightarrow y - a$ is the inverse isomorphism.

Clearly the relations $0 \leqslant x \leqslant b$ imply $a \leqslant a + x \leqslant a + b$. The mapping $x \rightarrow a + x$ is strictly increasing (and therefore injective) by Proposition 3 of no. 2. Finally, the relations $a \leqslant y \leqslant a + b$ imply $y = a + x$ with $x \geqslant 0$ and $a + x \leqslant a + b$, whence $x \leqslant b$ (no. 2, Proposition 3). This completes the proof.