Proposition 2

Let $a$ and $b$ be two integers. Then $a < b$ if and only if there exists an integer $c > 0$ such that $b = a + c$.

If $a < b$, there exists a cardinal $c \leqslant b$ (so that $c$ is an integer (§ 4, no. 2, Proposition 2)) such that $b = a + c$ (§ 3, no. 6, Proposition 13); if $a \neq b$, we must have $c \neq 0$. Conversely, if $b = a + c$ and $c \neq 0$, then $c \geqslant 1$ and hence $a < a + 1 \leqslant a + c = b$.