\begin{align*}
0 \longrightarrow E' \xrightarrow{\iota} E \xrightarrow{\pi} E'' \longrightarrow 0
\end{align*}
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be a short exact sequence of holomorphic vector bundles over $X$, regarded also as smooth complex vector bundles by forgetting the holomorphic structure. For every holomorphic vector bundle $F \to X$, let
denote its total Chern class, defined by the standard Chern-Weil normalization from any smooth connection on the underlying smooth complex vector bundle. Then
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\begin{align*}
c(E)=c(E')\,c(E'')
\end{align*}
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in the graded ring $H_{\mathrm{dR}}^{2\bullet}(X;\mathbb{C})$.