(The following is crossposted from [Math.SE][1], where the question did not receive any answers.)

I am looking for a proof of the following lemma from P. Gabriel's *Des catégories abéliennes* (Chap. IV, §3, Lemme 1):

> **Lemma.** Let $B$ be a ring, $(I, \leq)$ be a directed poset, and $(M_i, f_{ji}:M_j\rightarrow M_i)_{j\geq i},$ $(N_i, g_{ji}:N_j\rightarrow N_i)_{j\geq i}$ two inverse systems of $B$-modules. Let $(h_i: M_i\rightarrow N_i)_i$ be a morphism of the inverse systems, and assume that all the maps $h_i$ are surjective with an Artinian kernel. Then the limit map
$$\varprojlim_i h_i: \varprojlim_i M_i\rightarrow \varprojlim_i N_i$$ is surjective.

I am looking either for a proof (I haven't manage to come up with one so far), or for a reference to a proof - Gabriel refers to "Bourbaki, *Topologie*, I Appendice, $3^{\text{e}}$ éd.", which is a reference I cannot find anywhere.

I do not want to assume that e.g. $I$ is countable - the reason is that I need this lemma, similarly as Gabriel in his thesis, to establish some properties of pseudo-compact modules over a pseudo-compact ring (namely that quotient of a pseudo-compact module by a closed submodule is pseudo-compact, and exactness of inverse limits). For this reason, as far as I can tell, I cannot use countable index sets (i.e. countable bases of neighbourhoods) in general.

Thank you in advance for any help. 

  [1]: https://math.stackexchange.com/questions/2759503/surjectivity-of-a-map-on-inverse-limits