The following theorem is often called the fundamental theorem of projective geometry:

Let $k$ be a field and let $n \geq 3$. Let $X$ be the partially ordered set of nonzero proper subspaces of $k^n$. Then every poset automorphism of $X$ is induced by a semi-linear automorphism of $k^n$, i.e. a set map $f:k^n\rightarrow k^n$ for which there exists a field automorphism $\tau:k \rightarrow k$ such that $$f(c_1 \vec{v}_1+c_2\vec{v}_2) = \tau(c_1) f(\vec{v}_1) + \tau(c_2) f(\vec{v}_2)$$ for all $c_1,c_2 \in k$ and $\vec{v}_1,\vec{v}_2 \in k^n$.

Question: who was the first person to prove this, and where does their proof appear? I know it has its origins in 19th century work of von Staudt, but I don't think that the above theorem appears in his work. On page 52 of Baer's book "Linear Algebra and Projective Geometry", he says that the first proof was due to Kamke, but he does not give a reference.