Proof of fundamental theorem of affine geometry using fund. thm. of projective geometry

Question

I am looking for a way to prove the fundamental theorem of affine geometry using the fundamental theorem of projective geometry.

For concreteness, let $V=K^n$ be the $n$-dimensional coordinate vector space over a field $K$ and denote by $(P(V),\subseteq)$ and $(AP(V),\subseteq)$ the posets of linear and affine subspaces of $V$ ordered by set inclusion. Then,

(Fund. thm. of projective geometry) (c.f. Theorem 17 in [1])

Any poset automorphism of $P(V)$ is induced by a semi-linear bijection on $V$.

and

(Fund. thm. of affine geometry) (c.f. Theorem 15 in [1])

Poset automorphisms of $AP(V)$ are in one-to-one correspondence with semi-affine bijections on $V$.

The book [1] proves the projective theorem using the affine one. However, in [2 section 3.1] I saw an idea of how to prove the reverse direction. In essence, the idea was to use the canonical embedding of $K^n$ into $PK^{n}$, i.e. $PK^n=PK^{n-1}\cup K^n$, to identify affine lines in $K^n$ with projective lines in $PK^n$. I assume that now one best uses a reformulation of the fund. projective theorem

(Fund. thm. of projective geometry) (c.f. Theorem 2.26 in [3])

Any bijection $\sigma:PK^n\rightarrow PK^n$ that takes any three collinear points into collinear points is induced by a semi-linear bijection on $K^{n+1}$.

I guess(*) that automorphisms of $AP(V)$ induce such bijections $\sigma$ of $PK^n$ and that(**) there is some kind of correspondence between semi-linear maps on $K^{n+1}$ and semi-affine maps on $K^n$ that allows to conclude the affine theorem.

My questions:

1. Have you seen such a full proof in the literature (using the idea outlined below or some other idea)?
2. Are my guesses * and ** correct, and how would you go about making them precise?

I'm looking into these things to get ideas for a similar proof in a symplectic setting, tying to build on a 'fund. thm. of symplectic geometry' I found in 4, but I'm not a mathematician so I'd be very happy about your insights!

[1]: Bennett, M. K. (2011). Affine and projective geometry.

[2]: arXiv:1604.01762 [math.GM]

[3]: Artin, E. (2016). Geometric algebra.