Lemma 2

Let I be a right filtered ordered set, $(E_\alpha, \varphi_{\beta\alpha})$ an inductive system of sets relative to I, E its inductive limit and $\varphi_\alpha : E_\alpha \to E, \alpha \in I$, the canonical mappings. Let $f : I \to I$ be a mapping such that $f(\alpha) > \alpha$ for $\alpha \in I$, and suppose given, for each $\alpha \in I$, a set $L_\alpha$ and mappings $u_\alpha : E_\alpha \to L_\alpha$ and $v_\alpha : L_\alpha \to E_{f(\alpha)}$ such that $v_\alpha \circ u'_\alpha = \varphi_{f(\alpha), \alpha}$. Let J be the ordered set obtained by providing I with the relation « $\alpha \leq \beta$ if $\alpha = \beta$ or $f(\alpha) \leq \beta$ ». If $\alpha, \beta \in J$ with $\alpha \leq \beta$, let $\psi_{\beta\alpha} : L_\alpha \to L_\beta$ be the mapping such that $\psi_{\beta\alpha} = \mathrm{Id}$ if $\alpha = \beta$, $\psi_{\beta\alpha} = u_\beta \circ \varphi_{\beta, f(\alpha)} \circ v_\alpha$ if $f(\alpha) \leq \beta$. If $\alpha \in J$, let $\psi_\alpha : L_\alpha \to E$ be the mapping $\varphi_{f(\alpha)} \circ v_\alpha$. Then the ordered set J is filtered, $(L_\alpha, \psi_{\beta\alpha})$ is an inductive system relative to J, $(\psi_\alpha)$ is an inductive system of mappings and the mapping $\psi : \lim_{\alpha \in J} L_\alpha \to E$ deduced from the $\psi_\alpha$ is bijective.

It is clear that J is inductive. If $\alpha, \beta \in J$ with $\alpha < \beta$, we have

analogously, if $\alpha, \beta, \gamma \in J$ with $\alpha < \beta < \gamma$, we have Let us prove the last assertion: for each $\alpha \in J$, we have hence $\varphi_\alpha(E_\alpha) = \psi_\alpha(u_\alpha(E_\alpha)) \subset \psi_\alpha(L_\alpha)$, and $\psi$ is surjective. Let $\alpha \in J$ and let $x, y \in L_\alpha$ with $\psi_\alpha(x) = \psi_\alpha(y)$, i.e. $\varphi_{f(\alpha)}(v_\alpha(x)) = \varphi_{f(\alpha)}(v_\alpha(y))$; there exists $\beta \in I$, $\beta \geq f(\alpha)$ such that hence and $\psi$ is injective.