Fundamental Theorem of real projective geometry: Let $f\colon \RP^n \to \RP^n$, $n \ge 2$, be a bijective map that maps lines to lines. Then $f$ is a projective transformation.
Remark: The theorem does not hold for arbitrary fields. For example $f\colon \CP^n \to \CP^n$ ($n \ge 2$) with
\[f\left(\sqvector{z_1\\ \vdots\\ z_{n+1}}\right) = \sqvector{\bar z_1\\ \vdots\\ \bar z_{n+1}}\]
is a bijective map, mapping complex projective lines to complex projective lines but it is not a projective transformation of $\CP^n$.
