I want to prove that a projective image from a projective line $k$ to the same projective line $k$ is a composition of projections.
I've seen a proof for the case that the image maps from $k$ to another projective line $m$. Here, three points $P$, $Q$, $R$ are introduced on $k$ with corresponding images $P'$, $Q'$, $R'$ on $m$. Then, this proof uses the intersection points of $PQ'$ with $P'Q$ and $RS'$ with $R'S$ to project $k$ from $Q'$ on the line between those intersection points and subsequently this line between the intersection points on $m$ from $Q$. However, if the image maps from $k$ to $k$, I don't have those intersection points...
Is there a possibility to adjust the proof to make it work for the image that maps from $k$ to $k$? I've already received a hint to introduce a new line, but I'm not sure how to use this.
To map from $k$ to $k$, take any line $m$ and point $O$. Then project $k$ from $O$ onto $m$, taking $P,Q,R$ to $P'',Q'',R''$. You already know how to map points $P'',Q'',R''$ on $m$ to points $P',Q',R'$ on $k$ via two projections. So now you can get a map from $P,Q,R$ to $P',Q',R'$ as a composition of all three projections.