For the proof of a certain combinatorial statement on subsets of ${\mathbb F}\_2^n$
in a paper I and several other people are working on, the following statement was helpful.


**Proposition.**
Let $V$ be a vector space over a field $F$, let $I$ be a finite set, and let
$U\_i \subseteq V$ for $i \in I$ be linear subspaces such that
$\dim \sum\_{j \in J} U\_j \ge \#J$ for each subset $J \subseteq I$.
Then there exists a linearly independent family $(v\_i) \in \prod\_{i \in I} U\_i$.


I have a proof for this, but I also have the feeling that this should be known.


So my question is: Has somebody seen this somewhere, and if so, can you give
a reference?




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**EDIT:** Following up on Fedor Petrov's answer, I found the original
reference:


R. Rado, *A theorem on independence relations*, [Quart. J. Math. Oxford Ser. **13** (1942), 83-89](https://doi.org/10.1093/qmath/os-13.1.83).


The statement I was looking for is Theorem 1 there.



Linearly independent vectors from a family of subspaces
This is exactly Rado theorem on independent transversal (for the corresponding linear matroid).


Strictly speaking, to use it you should deal with finite sets, but a dimension of a vector subspace always equals to a rank of its appropriate finite subset, so there is no problem with it.


