We prove this lemma by induction on the number of generators of $I$. Say $I = (a_1, \ldots, a_t)$. If $t = 0$, then $I = 0$ and there is nothing to prove. If $t = 1$, then the lemma follows from Lemma \ref{lemma-approximate-principal}. Assume $t > 1$. \medskip\noindent For any $m \geq 1$ set $\bar A_m = A/(a_t^m)$. Consider the ideal $\bar I_m = (\bar a_1, \ldots, \bar a_{t - 1})$ in $\bar A_m$. Observe that $V(I \bar A_m) = V(\bar I_m)$. Let $B_m = B/(a_t^m)$ be the base change of $B$ for the map $(A, I) \to (\bar A_m, \bar I_m)$, see Remark \ref{remark-take-bar}. By Lemma \ref{lemma-base-change-rig-etale-homomorphism} we find that $B_m$ is rig-\'etale over $(\bar A_m, \bar I_m)$. \medskip\noindent By induction hypothesis (on $t$) we can find a finite type $\bar A_m$-algebra $C_m$ and a map $C_m \to B_m$ which induces an isomorphism $C_m^\wedge \cong B_m$ where the completion is with respect to $\bar I_m$. By Lemma \ref{lemma-rig-etale} we may assume that $\Spec(C_m) \to \Spec(\bar A_m)$ is \'etale over $\Spec(\bar A_m) \setminus V(\bar I_m)$. \medskip\noindent We claim that we may choose $A_m \to C_m \to B_m$ as in the previous paragraph such that moreover there are isomorphisms $C_m/(a_t^{m - 1}) \to C_{m - 1}$ compatible with the given $A$-algebra structure and the maps to $B_{m - 1} = B_m/(a_t^{m - 1})$. Namely, first fix a choice of $A_1 \to C_1 \to B_1$. Suppose we have found $C_{m - 1} \to C_{m - 2} \to \ldots \to C_1$ with the desired properties. Note that $C_m/(a_t^{m - 1})$ is \'etale over $\Spec(\bar A_{m - 1}) \setminus V(\bar I_{m - 1})$. Hence by Lemma \ref{lemma-fully-faithful-etale-over-complement} there exists an \'etale extension $C_{m - 1} \to C'_{m - 1}$ which induces an isomorphism modulo $\bar I_{m - 1}$ and an $\bar A_{m - 1}$-algebra map $C_m/(a_t^{m - 1}) \to C'_{m - 1}$ inducing the isomorphism $B_m/(a_t^{m - 1}) \to B_{m - 1}$ on completions. Note that $C_m/(a_t^{m - 1}) \to C'_{m - 1}$ is \'etale over the complement of $V(\bar I_{m - 1})$ by Morphisms, Lemma \ref{morphisms-lemma-etale-permanence} and over $V(\bar I_{m - 1})$ induces an isomorphism on completions hence is \'etale there too (for example by More on Morphisms, Lemma \ref{more-morphisms-lemma-check-smoothness-on-infinitesimal-nbhds}). Thus $C_m/(a_t^{m - 1}) \to C'_{m - 1}$ is \'etale. By the topological invariance of \'etale morphisms (\'Etale Morphisms, Theorem \ref{etale-theorem-remarkable-equivalence}) there exists an \'etale ring map $C_m \to C'_m$ such that $C_m/(a_t^{m - 1}) \to C'_{m - 1}$ is isomorphic to $C_m/(a_t^{m - 1}) \to C'_m/(a_t^{m - 1})$. Observe that the $\bar I_m$-adic completion of $C'_m$ is equal to the $\bar I_m$-adic completion of $C_m$, i.e., to $B_m$ (details omitted). We apply Lemma \ref{lemma-lift-approximation} to the diagram $$ \xymatrix{  & C'_m \ar[r] & C'_m/(a_t^{m - 1}) \\ C''_m \ar@{..>}[ru] \ar@{..>}[rr] & & C_{m - 1} \ar[u] \\  & \bar A_m \ar[r] \ar[uu] \ar@{..>}[lu] & \bar A_{m - 1} \ar[u] } $$ to see that there exists a ``lift'' of $C''_m$ of $C_{m - 1}$ to an algebra over $\bar A_m$ with all the desired properties. \medskip\noindent By construction $(C_m)$ is an object of the category (\ref{equation-C}) for the principal ideal $(a_t)$. Thus the inverse limit $B' = \lim C_m$ is an $(a_t)$-adically complete $A$-algebra such that $B'/a_t B'$ is of finite type over $A/(a_t)$, see Lemma \ref{lemma-topologically-finite-type}. By construction the $I$-adic completion of $B'$ is isomorphic to $B$ (details omitted). Consider the complex $\NL_{B'/A}^\wedge$ constructed using the $(a_t)$-adic topology. Choosing a presentation for $B'$ (which induces a similar presentation for $B$) the reader immediately sees that $\NL_{B'/A}^\wedge \otimes_{B'} B = \NL_{B/A}^\wedge$. Since $a_t \in I$ and since the cohomology modules of $\NL_{B'/A}^\wedge$ are finite $B'$-modules (hence complete for the $a_t$-adic topology), we conclude that $a_t^c$ acts as zero on these cohomologies as the same thing is true by assumption for $\NL_{B/A}^\wedge$. Thus $B'$ is rig-\'etale over $(A, (a_t))$ by Lemma \ref{lemma-equivalent-with-artin}. Hence finally, we may apply Lemma \ref{lemma-approximate-principal} to $B'$ over $(A, (a_t))$ to finish the proof.