Let $$ 0 \to \mathcal{K} \to \mathcal{F} \to \mathcal{F}' \to 0 $$ be the short exact sequence constructed in Divisors, Lemma \ref{divisors-lemma-remove-embedded-points}. This in particular means that $\mathcal{F}'$ has no embedded associated points. Since the support of $\mathcal{K}$ is nowhere dense in the support of $\mathcal{F}$ we see that $\dim_\delta(\text{Supp}(\mathcal{K})) \leq k$. We may re-apply Divisors, Lemma \ref{divisors-lemma-remove-embedded-points} starting with $\mathcal{K}$ to get a short exact sequence $$ 0 \to \mathcal{K}'' \to \mathcal{K} \to \mathcal{K}' \to 0 $$ where now $\dim_\delta(\text{Supp}(\mathcal{K}'')) < k$ and $\mathcal{K}'$ has no embedded associated points. Suppose we can prove the lemma for the coherent sheaves $\mathcal{F}'$ and $\mathcal{K}'$. Then we see from the equations $$ [\mathcal{F}]_{k + 1} = [\mathcal{F}']_{k + 1} + [\mathcal{K}']_{k + 1} + [\mathcal{K}'']_{k + 1} $$ (use Lemma \ref{lemma-additivity-sheaf-cycle}), $$ [\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}] - [\mathcal{F}] = [\mathcal{F}' \otimes_{\mathcal{O}_X} \mathcal{L}] - [\mathcal{F}'] + [\mathcal{K}' \otimes_{\mathcal{O}_X} \mathcal{L}] - [\mathcal{K}'] + [\mathcal{K}'' \otimes_{\mathcal{O}_X} \mathcal{L}] - [\mathcal{K}''] $$ (use the $\otimes \mathcal{L}$ is exact) and the trivial vanishing of $[\mathcal{K}'']_{k + 1}$ and $[\mathcal{K}'' \otimes_{\mathcal{O}_X} \mathcal{L}] - [\mathcal{K}'']$ in $K_0(\textit{Coh}_{\leq k + 1}(X)/\textit{Coh}_{\leq k - 1}(X))$ that the result holds for $\mathcal{F}$. What this means is that we may assume that the sheaf $\mathcal{F}$ has no embedded associated points. \medskip\noindent Assume $X$, $\mathcal{F}$ as in the lemma, and assume in addition that $\mathcal{F}$ has no embedded associated points. Consider the sheaf of ideals $\mathcal{I} \subset \mathcal{O}_X$, the corresponding closed subscheme $i : Z \to X$ and the coherent $\mathcal{O}_Z$-module $\mathcal{G}$ constructed in Divisors, Lemma \ref{divisors-lemma-no-embedded-points-endos}. Recall that $Z$ is a locally Noetherian scheme without embedded points, $\mathcal{G}$ is a coherent sheaf without embedded associated points, with $\text{Supp}(\mathcal{G}) = Z$ and such that $i_*\mathcal{G} = \mathcal{F}$. Moreover, set $\mathcal{N} = \mathcal{L}|_Z$. \medskip\noindent By Divisors, Lemma \ref{divisors-lemma-regular-meromorphic-section-exists} the invertible sheaf $\mathcal{N}$ has a regular meromorphic section $s$ over $Z$. Let us denote $\mathcal{J} \subset \mathcal{O}_Z$ the sheaf of denominators of $s$. By Lemma \ref{lemma-no-embedded-points} there exist short exact sequences $$ \begin{matrix} 0 & \to & \mathcal{J}\mathcal{G} & \xrightarrow{1} & \mathcal{G} & \to & \mathcal{Q}_1 & \to & 0 \\ 0 & \to & \mathcal{J}\mathcal{G} & \xrightarrow{s} & \mathcal{G} \otimes_{\mathcal{O}_Z} \mathcal{N} & \to & \mathcal{Q}_2 & \to & 0 \end{matrix} $$ such that $\dim_\delta(\text{Supp}(\mathcal{Q}_i)) \leq k$ and such that the cycle $ [\mathcal{Q}_2]_k - [\mathcal{Q}_1]_k $ is a representative of $c_1(\mathcal{N}) \cap [\mathcal{G}]_{k + 1}$. We see (using the fact that $i_*(\mathcal{G} \otimes \mathcal{N}) = \mathcal{F} \otimes \mathcal{L}$ by the projection formula, see Cohomology, Lemma \ref{cohomology-lemma-projection-formula}) that $$ [\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}] - [\mathcal{F}] = [i_*\mathcal{Q}_2] - [i_*\mathcal{Q}_1] $$ in $K_0(\textit{Coh}_{\leq k + 1}(X)/\textit{Coh}_{\leq k - 1}(X))$. This already shows that $[\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}] - [\mathcal{F}]$ is an element of $B_k(X)$. Moreover we have \begin{eqnarray*} [i_*\mathcal{Q}_2]_k - [i_*\mathcal{Q}_1]_k & = & i_*\left( [\mathcal{Q}_2]_k - [\mathcal{Q}_1]_k \right) \\ & = & i_*\left(c_1(\mathcal{N}) \cap [\mathcal{G}]_{k + 1} \right) \\ & = & c_1(\mathcal{L}) \cap i_*[\mathcal{G}]_{k + 1} \\ & = & c_1(\mathcal{L}) \cap [\mathcal{F}]_{k + 1} \end{eqnarray*} by the above and Lemmas \ref{lemma-pushforward-cap-c1} and \ref{lemma-cycle-push-sheaf}. And this agree with the image of the element under $B_k(X) \to \CH_k(X)$ by definition. Hence the lemma is proved.