Lemma 1

Every infinite set E contains a set equipotent to $\mathbf{N}$.

There exists a well-ordering relation on E (§ 2, no. 3, Theorem 1), which we shall denote by $x \leqslant y$. The hypothesis implies that the well-ordered set E cannot be isomorphic to a segment of $\mathbf{N}$ distinct from $\mathbf{N}$, for such a segment is of the form $[0, n]$ (§ 2, no. 1, Proposition 1) and is therefore finite. It follows that $\mathbf{N}$ is isomorphic to a segment of E (§ 2, no. 3, Theorem 3), whence the result.