Choose a $\alpha : \Lambda[x_1, \ldots, x_n] \to A$ with kernel $J$. Write $P = \Lambda[x_1, \ldots, x_n]$. In the rest of this proof we work with $$ \NL(\alpha) = (J/J^2 \longrightarrow \bigoplus A \text{d}x_i) $$ which is permissible by Algebra, Lemma \ref{algebra-lemma-NL-homotopy} and More on Algebra, Lemma \ref{more-algebra-lemma-derived-tor-homotopy}. Consider the element $o_x(P/J^2) \in H^2(K^\bullet \otimes_A^\mathbf{L} J/J^2)$ and consider the quotient $$ C = (P/J^2 \times \bigoplus A \text{d}x_i)/(J/J^2) $$ where $J/J^2$ is embedded diagonally. Note that $C \to A$ is a surjection with kernel $\bigoplus A\text{d}x_i$. Moreover there is a section $A \to C$ to $C \to A$ given by mapping the class of $f \in P$ to the class of $(f, \text{d}f)$ in the pushout. For later use, denote $x_C$ the pullback of $x$ along the corresponding morphism $\Spec(C) \to \Spec(A)$. Thus we see that $o_x(C) = 0$. We conclude that $o_x(P/J^2)$ maps to zero in $H^2(K^\bullet \otimes_A^\mathbf{L} \bigoplus A\text{d}x_i)$. It follows that there exists some element $\xi \in H^1(K^\bullet \otimes_A^\mathbf{L} \NL(\alpha))$ mapping to $o_x(P/J^2)$. \medskip\noindent Note that for any deformation situation $(x, A' \to A)$ there exists a $\Lambda$-algebra map $P/J^2 \to A'$ compatible with the augmentations to $A$. Hence the element $\xi$ satisfies the first property of the lemma by construction and property (ii) of Situation \ref{situation-dual}. \medskip\noindent Note that our choice of $\xi$ was well defined up to the choice of an element of $H^1(K^\bullet \otimes_A^\mathbf{L} \bigoplus A\text{d}x_i)$. We will show that after modifying $\xi$ by an element of the aforementioned group we can arrange it so that the second assertion of the lemma is true. Let $C' \subset C$ be the image of $P/J^2$ under the $\Lambda$-algebra map $P/J^2 \to C$ (inclusion of first factor). Observe that $\Ker(C' \to A) = \Im(J/J^2 \to \bigoplus A\text{d}x_i)$. Set $\overline{C} = A[\Omega_{A/\Lambda}]$. The map $P/J^2 \times \bigoplus A \text{d}x_i \to \overline{C}$, $(f, \sum f_i \text{d}x_i) \mapsto (f \bmod J, \sum f_i \text{d}x_i)$ factors through a surjective map $C \to \overline{C}$. Then $$ (x, \overline{C} \to A) \to (x, C \to A) \to (x, C' \to A) $$ is a short exact sequence of deformation situations. The associated splitting $\overline{C} = A[\Omega_{A/\Lambda}]$ (from Remark \ref{remark-short-exact-sequence-thickenings}) equals the given splitting above. Moreover, the section $A \to C$ composed with the map $C \to \overline{C}$ is the map $(1, \text{d}) : A \to A[\Omega_{A/\Lambda}]$ of Remark \ref{remark-canonical-element}. Thus $x_C$ restricts to the canonical element $x_{can}$ of $T_x(\Omega_{A/\Lambda}) = \text{Lift}(x, A[\Omega_{A/\Lambda}])$. By condition (iv) we conclude that $o_x(P/J^2)$ maps to $\partial x_{can}$ in $$ H^1(K^\bullet \otimes_A^\mathbf{L} \Im(J/J^2 \to \bigoplus A\text{d}x_i)) $$ By construction $\xi$ maps to $o_x(P/J^2)$. It follows that $x_{can}$ and $\xi_{can}$ map to the same element in the displayed group which means (by the long exact cohomology sequence) that they differ by an element of $H^1(K^\bullet \otimes_A^\mathbf{L} \bigoplus A\text{d}x_i)$ as desired.