Lemma 1

Let $(\mathrm{E}_\alpha, f_{\beta\alpha})$ be a direct system of sets, $\mathrm{E} = \varinjlim \mathrm{E}_\alpha$ its direct limit, and for each $\alpha \in \mathrm{I}$ let $f_\alpha \colon \mathrm{E}_\alpha \to \mathrm{E}$ the canonical mapping.

(i) Let $(x^{(i)})_{1 \leqslant i \leqslant n}$ be a finite system of elements of E. Then there exists $\alpha \in I$ and a finite system $(x_\alpha^{(i)})_{1 \leqslant i \leqslant n}$ of elements of $E_\alpha$ such that $x^{(i)} = f_\alpha(x_\alpha^{(i)})$ for $1 \leqslant i \leqslant n$.

(ii) Let $(y_\alpha^{(i)})_{1 \leqslant i \leqslant n}$ be a finite system of elements of some $E_\alpha$. If $f_\alpha(y_\alpha^{(i)}) = f_\alpha(y_\alpha^{(j)})$ for each pair of indices $(i, j)$, then there exists $\beta \geqslant \alpha$ such that $f_{\beta\alpha}(y_\alpha^{(i)}) = f_{\beta\alpha}(y_\alpha^{(j)})$ for each pair $(i, j)$.

(i) By the definition of E there exists for each $i$ an index $\beta_i \in I$ and an element $z_{\beta_i} \in E_{\beta_i}$ such that $x^{(i)} = f_{\beta_i}(z_{\beta_i})$. Take $\alpha$ such that $\alpha \geqslant \beta_i$ for $1 \leqslant i \leqslant n$, and $x_\alpha^{(i)} = f_{\alpha\beta_i}(z_{\beta_i})$.

(ii) By the definition of E, for each pair $(i, j)$ there exists $\gamma_{ij} \in I$ such that $\gamma_{ij} \geqslant \alpha$ and $f_{\gamma_{ij}\alpha}(y_\alpha^{(i)}) = f_{\gamma_{ij}\alpha}(y_\alpha^{(j)})$. Take $\beta$ such that $\beta \geqslant \gamma_{ij}$ for all pairs $(i, j)$, and use the relations $f_{\beta\alpha} = f_{\beta\gamma_{ij}} \circ f_{\gamma_{ij}\alpha}$.