  A subbundle of a Hermitian vector bundle $(E, h)$ can be metrically and
differentiably defined by the orthogonal projection onto this subbundle. A
weakly holomorphic subbundle of a Hermitian holomorphic bundle is, by
definition, an orthogonal projection $\pi$ lying in the Sobolev space $L^2_1$
of $L^2$ sections with $L^2$ first order derivatives in the sense of
distributions, which satisfies furthermore $(\mathrm{Id}-\pi)\circ D''\pi=0$.
We give a new simple proof of the fact that a weakly holomorphic subbundle of
$(E, h)$ defines a coherent subsheaf of ${\cal O}(E),$ that is a holomorphic
subbundle of $E$ in the complement of an analytic set of codimension $\geq 2.$
This result was the crucial technical argument in Uhlenbeck's and Yau's proof
of the Kobayashi-Hitchin correspondence on compact K\"ahler manifolds. We give
here a much simpler proof based on current theory. The idea is to construct
local meromorphic sections of $\mathrm{Im} \pi$ which locally span the fibers.
We first make this construction on every one-dimensional submanifold of $X$ and
subsequently extend it via a Hartogs-type theorem of Shiffman's.
