"Let A,B,C,D be four points on a projective line. Show that there's a projective transformation which switches places of A and C, and also B and D."
My thinking was that if I have a known projective line where I have a known projective transformation that does the switch, then I could just have that all projective lines can be mapped to (with inverse) any other projective line, and then the switch property would be true for every line, since I can map it to the known line, apply the 'switch' projective transformation, and then use the inverse map to go back to the original line.
The particular line and transformation that I thought of is x+y-z=0 and (x,y,z) -> (y,x,z).
Is my thinking correct? Does this prove it?