Corollary 1

Let $(G_\alpha, f_{\alpha \beta})$ be an inverse system of topological groups which satisfies the hypotheses of Proposition 5; let $(G'_\alpha, f'_{\alpha \beta})$ be an inverse system of topological groups, and for each $\alpha$ let $u_\alpha : G_\alpha \to G'_\alpha$ be a surjective strict morphism with compact kernel, such that the $u_\alpha$ from an inverse system of mappings. Then $u = \varprojlim u_\alpha$ is a strict morphism of $G = \varprojlim G_\alpha$ onto $G' = \varprojlim G'_\alpha$, and its kernel is compact.

Let $N_\alpha$ be the kernel of $u_\alpha$. $L_\alpha = f_\alpha^{-1}(N_\alpha)$ is then the kernel of the surjective strict morphism $v_\alpha = u_\alpha \circ f_\alpha : G \to G'_\alpha$; since $L_\alpha / H_\alpha$ is isomorphic to $N_\alpha$ (§ 2, no. 7, Proposition 20), $L_\alpha$ is a compact normal subgroup of $G$ (§ 4, no. 1, Corollary 2 to Proposition 2). The kernel $L$ of $u$ is the intersection of the $L_\alpha$. Let $\varphi$ denote the canonical mapping $G \to G/L$; then we can write $v_\alpha = w_\alpha \circ \varphi$, where $w_\alpha$ is a strict morphism of $G/L$ onto $G'_\alpha$, whose kernel is $L_\alpha / L$. Since the intersection of the $L_\alpha / L$ is the identity element of $G/L$, and since the $L_\alpha / L$ form a filter base and are compact, this filter base converges to the identity element of $G/L$ (Chapter I, § 9, no. 1, Corollary to Theorem 1). Proposition 2 of no. 3 then shows that $w = \varprojlim w_\alpha$ is an isomorphism of $G/L$ onto $G'$; it follows that $w \circ \varphi$ is a strict morphism of $G$ onto $G'$, with kernel $L$. But it is clear that $u = w \circ \varphi$, and thus the corollary is proved.