Say $Y = \Spec(A)$ and $S = \Spec(R)$. In this case $\Omega^\bullet_{A/R}$ computes $R\Gamma(Y, \Omega^\bullet_{Y/S})$ by Lemma \ref{lemma-de-rham-affine}. Choose a finite affine open covering $\mathcal{U} : X = \bigcup_{i \in I} U_i$. Consider the complex $$ K^\bullet = \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \Omega_{X/S}^\bullet)) $$ as in Cohomology, Section \ref{cohomology-section-cech-cohomology-of-complexes}. Let us collect some facts about this complex most of which can be found in the reference just given: \begin{enumerate} \item $K^\bullet$ is a complex of $R$-modules whose terms are $A$-modules, \item $K^\bullet$ represents $R\Gamma(X, \Omega^\bullet_{X/S})$ in $D(R)$ (Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherent-affine-cohomology-zero} and Cohomology, Lemma \ref{cohomology-lemma-cech-complex-complex-computes}), \item there is a natural map $\Omega^\bullet_{A/R} \to K^\bullet$ of complexes of $R$-modules which is $A$-linear on terms and induces the pullback map $H^*_{dR}(Y/S) \to H^*_{dR}(X/S)$ on cohomology, \item $K^\bullet$ has a multiplication denoted $\wedge$ which turns it into a differential graded $R$-algebra, \item the multiplication on $K^\bullet$ induces the cup product on $H^*_{dR}(X/S)$ (Cohomology, Section \ref{cohomology-section-cup-product}), \item the filtration $F$ on $\Omega^*_{X/S}$ induces a filtration $$ K^\bullet = F^0K^\bullet \supset F^1K^\bullet \supset F^2K^\bullet \supset \ldots $$ by subcomplexes on $K^\bullet$ such that \begin{enumerate} \item $F^kK^n \subset K^n$ is an $A$-submmodule, \item $F^kK^\bullet \wedge F^lK^\bullet \subset F^{k + l}K^\bullet$, \item $\text{gr}^kK^\bullet$ is a complex of $A$-modules, \item $\text{gr}^0K^\bullet = \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \Omega_{X/Y}^\bullet))$ and represents $R\Gamma(X, \Omega^\bullet_{X/Y})$ in $D(A)$, \item multiplication induces an isomorphism $\Omega^k_{A/R}[-k] \otimes_A \text{gr}^0K^\bullet \to \text{gr}^kK^\bullet$ \end{enumerate} \end{enumerate} We omit the detailed proofs of these statements; please see discussion leading up to the construction of the spectral sequence in Lemma \ref{lemma-spectral-sequence-smooth}. \medskip\noindent For every $i = 1, \ldots, N$ we choose a cocycle $x_i \in K^{n_i}$ representing $\xi_i$. Next, we look at the map of complexes $$ \tilde x : M^\bullet = \bigoplus\nolimits_{i = 1, \ldots, N} \Omega^\bullet_{A/R}[-n_i] \longrightarrow K^\bullet $$ which sends $\omega$ in the $i$th summand to $x_i \wedge \omega$. All that remains is to show that this map is a quasi-isomorphism. We endow $M^\bullet$ with the structure of a filtered complex by the rule $$ F^kM^\bullet = \bigoplus\nolimits_{i = 1, \ldots, N} (\sigma_{\geq k}\Omega^\bullet_{A/R})[-n_i] $$ With this choice the map $\tilde x$ is a morphism of filtered complexes. Observe that $\text{gr}^0M^\bullet = \bigoplus A[-n_i]$ and multiplication induces an isomorphism $\Omega^k_{A/R}[-k] \otimes_A \text{gr}^0M^\bullet \to \text{gr}^kM^\bullet$. By construction and Lemma \ref{lemma-relative-global-generation-on-fibres} we see that $$ \text{gr}^0\tilde x : \text{gr}^0M^\bullet \longrightarrow \text{gr}^0K^\bullet $$ is an isomorphism in $D(A)$. It follows that for all $k \geq 0$ we obtain isomorphisms $$ \text{gr}^k \tilde x : \text{gr}^kM^\bullet = \Omega^k_{A/R}[-k] \otimes_A \text{gr}^0M^\bullet \longrightarrow \Omega^k_{A/R}[-k] \otimes_A \text{gr}^0K^\bullet = \text{gr}^kK^\bullet $$ in $D(A)$. Namely, the complex $\text{gr}^0K^\bullet = \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \Omega_{X/Y}^\bullet))$ is K-flat as a complex of $A$-modules by Derived Categories of Schemes, Lemma \ref{perfect-lemma-K-flat}. Hence the tensor product on the right hand side is the derived tensor product as is true by inspection on the left hand side. Finally, taking the derived tensor product $\Omega^k_{A/R}[-k] \otimes_A^\mathbf{L} -$ is a functor on $D(A)$ and therefore sends isomorphisms to isomorphisms. Arguing by induction on $k$ we deduce that $$ \tilde x : M^\bullet/F^kM^\bullet \to K^\bullet/F^kK^\bullet $$ is an isomorphism in $D(R)$ since we have the short exact sequences $$ 0 \to F^kM^\bullet/F^{k + 1}M^\bullet \to M^\bullet/F^{k + 1}M^\bullet \to \text{gr}^kM^\bullet \to 0 $$ and similarly for $K^\bullet$. This proves that $\tilde x$ is a quasi-isomorphism as the filtrations are finite in any given degree.