How to show that the only affine automorphism of $\mathbb{R}$ is the identity, using fundamental thorem of affine geometry? Let the set of real numbers $\mathbb{R}$ be endowed with the natural affine structure (that means the group action is the usual addition). The question is how to prove that the only affine automorphism in affine space $\mathbb{R}$ is the identity, using the fundamental theorem of affine geometry. The affine automorphism is an affine map which is an automorphism. The fundamental theorem of affine geometry states as follows:

Let $X,X'$ be affine spaces of the same affine dimension $\ge2$. Let $f:X\to X'$ be a set-theoretical bijection which takes any three collinear points in $X$ into collinear points in $X'$. Then $f$ is semiaffine.

I have no idea know how to relate the fundamental theorem of affine geometry to the problem in question. In particular, the affine space $\mathbb{R}$ has affine dimension $1$, not $\ge2$ as stipulated in the theorem. What contradiction can I get if I assume there is a non-identity automorphism in affine space $\mathbb{R}$? I would appreciate your help greatly if you can direct me how to use this powerful fundamental theorem.
PS, for reference, the screenshot of the proposition and the fundamental theorem is copied below:

 A: The statements about $\mathbb{R}$ and $\mathbb{C}$ in the Proposition are about field automorphisms. The point is that

*

*It is well known that the only field automorphism of $\mathbb{R}$ is the identity (the fact that the state this is what tells you they are talking about field automorphisms, since there are many examples of affine bijections of the real number line that are not field automorphisms).

*We can view $\mathbb{C}$ as a 2-dimensional affine geometry over $\mathbb{R}$.

*We can use the fact that a "continuous automorphism" of $\mathbb{C}$ must be an affine automorphism, that is, it is both an affine map in the sense of the fundamental theorem of affine geometry, and also a field automorphism.

This lets you use the Fundamental Theorem (which is about affine maps) to classify the continuous field automorphisms of $\mathbb{C}$. Notice that there is no claim about the transformations of $\mathbb{R}$ as an affine geometry; indeed the Fundamental Theorem does not apply here because $\mathbb{R} = \mathbb{R}^{1}$ is one-dimensional.
