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This result is needed in our paper arXiv:1201.1672. The result presented here has also applications in other situations and so it should appear as part of a larger work.\n  The statement of the lemma goes as follows: Suppose $X$ is a nonempty algebraically closed subset of the affine space of $n \\times m$ complex matrices. Suppose that $X$ is column-invariant (i.e., belongingness to $X$ depends only on the column space of the matrix). 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