[proofplan] Theorem A follows from Theorem B by killing the obstruction to lifting a prescribed germ. We first reduce a germ modulo the maximal ideal and use a skyscraper quotient. The long exact sequence in cohomology then shows that the required local class is the germ of a global section. Nakayama's lemma upgrades generation modulo the maximal ideal to generation of the stalk as an $\mathcal{O}_{X,p}$-module. [/proofplan] [step:Form the quotient detecting the value of a germ at $p$] Let $\mathfrak{m}_p\subset\mathcal{O}_{X,p}$ be the maximal ideal and let $\mathfrak{m}_p\mathcal{F}$ denote the coherent subsheaf of sections whose germ at $p$ lies in $\mathfrak{m}_p\mathcal{F}_p$, with no restriction away from $p$. There is a short exact sequence of coherent sheaves \begin{align*} 0\to \mathfrak{m}_p\mathcal{F}\to \mathcal{F}\to \mathcal{F}_p/\mathfrak{m}_p\mathcal{F}_p\to 0, \end{align*} where the last term is a skyscraper sheaf supported at $p$. [/step] [step:Use Theorem B to lift residue classes globally] Taking global sections gives the beginning of the long exact sequence \begin{align*} H^0(X,\mathcal{F})\to H^0(X,\mathcal{F}_p/\mathfrak{m}_p\mathcal{F}_p)\to H^1(X,\mathfrak{m}_p\mathcal{F}). \end{align*} The sheaf $\mathfrak{m}_p\mathcal{F}$ is coherent, and $X$ is Stein, so Cartan's Theorem B gives $H^1(X,\mathfrak{m}_p\mathcal{F})=0$. Therefore the map \begin{align*} H^0(X,\mathcal{F})\to \mathcal{F}_p/\mathfrak{m}_p\mathcal{F}_p \end{align*} is surjective. [/step] [step:Apply Nakayama's lemma at the stalk] Choose global sections $s_1,\dots,s_r$ whose germs map to generators of the finite-dimensional vector space $\mathcal{F}_p/\mathfrak{m}_p\mathcal{F}_p$. Let $M\subseteq\mathcal{F}_p$ be the $\mathcal{O}_{X,p}$-submodule generated by the germs $(s_i)_p$. The quotient $\mathcal{F}_p/M$ is a finitely generated $\mathcal{O}_{X,p}$-module and its reduction modulo $\mathfrak{m}_p$ is zero. Nakayama's lemma gives $\mathcal{F}_p/M=0$, so the germs of global sections generate $\mathcal{F}_p$. [/step]