Let $X \in \Ob(\mathcal{C})$ and let $\mathcal{E}$ be a finite locally free $\mathcal{O}_X$-module. We first show how to define the rank $r^A(\mathcal{E}) \in A^0(X)$. Namely, let $X = \bigcup X_r$ be the decomposition into open and closed subschemes such that $\mathcal{E}|_{X_r}$ has constant rank $r$. Since $X$ is quasi-compact, this decomposition is finite, say $X = X_0 \amalg X_1 \amalg \ldots \amalg X_n$. Then $A(X) = A(X_0) \times A(X_1) \times \ldots \times A(X_n)$. Thus we can define $r^A(\mathcal{E}) = (0, 1, \ldots, n) \in A^0(X)$. \medskip\noindent Let $P_p(c_1, \ldots, c_p)$ be the polynomials constructed in Chow Homology, Example \ref{chow-example-power-sum}. Then we can define $$ ch^A(\mathcal{E}) = r^A(\mathcal{E}) + \sum\nolimits_{i \geq 1} (1/i!) P_i(c^A_1(\mathcal{E}), \ldots, c^A_i(\mathcal{E})) \in \prod\nolimits_{i \geq 0} A^i(X) $$ where $ci^A$ are the Chern classes of Proposition \ref{proposition-chern-class}. It follows immediately that we have property (2) and (3) of the lemma. \medskip\noindent We still have to show the following three statements \begin{enumerate} \item If $0 \to \mathcal{E}_1 \to \mathcal{E} \to \mathcal{E}_2 \to 0$ is a short exact sequence of finite locally free $\mathcal{O}_X$-modules on $X \in \Ob(\mathcal{C})$, then $ch^A(\mathcal{E}) = ch^A(\mathcal{E}_1) + ch^A(\mathcal{E}_2)$. \item If $\mathcal{E}_1$ and $\mathcal{E}_2 \to 0$ are finite locally free $\mathcal{O}_X$-modules on $X \in \Ob(\mathcal{C})$, then $ch^A(\mathcal{E}_1 \otimes \mathcal{E}_2) = ch^A(\mathcal{E}_1) ch^A(\mathcal{E}_2)$. \end{enumerate} Namely, the first will prove that $ch^A$ factors through $K_0(\textit{Vect}(X))$ and the first and the second will combined show that $ch^A$ is a ring map. \medskip\noindent To prove these statements we can reduce to the case where $\mathcal{E}_1$ and $\mathcal{E}_2$ have constant ranks $r_1$ and $r_2$. In this case the equalities in $A^0(X)$ are immediate. To prove the equalities in higher degrees, by Lemma \ref{lemma-splitting-principle} we may assume that $\mathcal{E}_1$ and $\mathcal{E}_2$ have filtrations whose graded pieces are invertible modules $\mathcal{L}_{1, j}$, $j = 1, \ldots, r_1$ and $\mathcal{L}_{2, j}$, $j = 1, \ldots, r_2$. Using the multiplicativity of Chern classes we get $$ c_i^A(\mathcal{E}_1) = s_i(c_1^A(\mathcal{L}_{1, 1}), \ldots, c_1^A(\mathcal{L}_{1, r_1})) $$ where $s_i$ is the $i$th elementary symmetric function as in Chow Homology, Example \ref{chow-example-power-sum}. Similarly for $c_i^A(\mathcal{E}_2)$. In case (1) we get $$ c_i^A(\mathcal{E}) = s_i(c_1^A(\mathcal{L}_{1, 1}), \ldots, c_1^A(\mathcal{L}_{1, r_1}), c_1^A(\mathcal{L}_{2, 1}), \ldots, c_1^A(\mathcal{L}_{2, r_2})) $$ and for case (2) we get $$ c_i^A(\mathcal{E}_1 \otimes \mathcal{E}_2) = s_i(c_1^A(\mathcal{L}_{1, 1}) + c_1^A(\mathcal{L}_{2, 1}), \ldots, c_1^A(\mathcal{L}_{1, r_1}) + c_1^A(\mathcal{L}_{2, r_2})) $$ By the definition of the polynomials $P_i$ we see that this means $$ P_i(c^A_1(\mathcal{E}_1), \ldots, c^A_i(\mathcal{E}_1)) = \sum\nolimits_{j = 1, \ldots, r_1} c_1^A(\mathcal{L}_{1, j})^i $$ and similarly for $\mathcal{E}_2$. In case (1) we have also $$ P_i(c^A_1(\mathcal{E}), \ldots, c^A_i(\mathcal{E})) = \sum\nolimits_{j = 1, \ldots, r_1} c_1^A(\mathcal{L}_{1, j})^i + \sum\nolimits_{j = 1, \ldots, r_2} c_1^A(\mathcal{L}_{2, j})^i $$ In case (2) we get accordingly $$ P_i(c^A_1(\mathcal{E}_1 \otimes \mathcal{E}_2), \ldots, c^A_i(\mathcal{E}_1 \otimes \mathcal{E}_2)) = \sum\nolimits_{j = 1, \ldots, r_1} \sum\nolimits_{j' = 1, \ldots, r_2} (c_1^A(\mathcal{L}_{1, j}) + c_1^A(\mathcal{L}_{2, j'}))^i $$ Thus the desired equalities are now consequences of elementary identities between symmetric polynomials. \medskip\noindent We omit the proof of uniqueness.