I found the following statement in P. Gabriel's thesis:
Lemma. Let $(I, \leq)$ be a directed poset, and $(M_i, \mu_{ji}:M_j\rightarrow M_i)_{j\geq i},$ $(N_i, \nu_{ji}:N_j\rightarrow N_i)_{j\geq i}$ two inverse systems of $R$-modules (let's say $R$ is a commutative ring). Let $(f_i: M_i\rightarrow N_i)_i$ be a system of compatible maps, and assume that all $f_i$'s are surjective and the kernels $K_i$ are Artinian modules. Then the limit map $$\varprojlim_i f_i: \varprojlim_i M_i\rightarrow \varprojlim_i N_i$$ is surjective.
The proof is not given, the reader is referred to an Appendix of Bourbaki's "topology" which I am unable to locate.
So my question is
How does one prove the lemma?
Some (maybe not-so useful) restatement: An element in $\varprojlim_i N_i$ can be viewed as a system of compatible elements $(n_i)_i$ in the usual way. Choosing the preimages $m_i \in M_i$ "at random", one can then check that for all $j \geq i$, $\mu_{ji}(m_j)-m_i \in K_i$. So any potential choice of preimages gives a collection $x=(\mu_{ji}(m_j)-m_i)_{j \geq i} \in \prod_{j\geq i}K_i$, and to get a compatible system is to be able to choose $m_i$'s such that $x=0$. I don't see, however, how $K_i$ being Artinian can help me with this. Can one, for example, make the sequences $(\mu_{j{i_0}}(m_j)-m_{i_0})_{j \geq i_0}$ for each fixed $i_0$ in a manner that the system of submodules $(\langle \mu_{j{i_0}}(m_j)-m_{i_0}\rangle)_{j \geq i_0}$ of $K_i$ is decreasing? If so, how does one do that? And also, would that ultimately help?
Let me also point out that $I$ should really be a general directed poset. In particular, I don't want to assume that $I$ is countable, $I$ is linear (unless there is wlog some linear cofinal chain, which I don't see why that would be the case).
Thanks in advance for any help. References are also welcome.