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Let $K$ be a field and $E$ an affine space over $K$. A subset $V$ of $E$ is an affine subspace (i.e. affine combinations of points of $V$ are in $V$) iff for any pair of distinct points of $V$ the line connecting them is in $V$.

I have been able to prove (by induction over the number of points in an affine combination) this in the case $\text{char}(K)\neq 2$. It is wrong for $K=F_2$ (in that case two distinct points already are a line).

My question: is it true for all other fields of characteristic 2?

My proof breaks down for affine combinations where all coefficients are equal to $1$, which may help in finding a counterexample. However, my intuition is very limited, and help is much appreciated.

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