I've worked through an example using the Fano Plane, and it seems to be the case. However, I'm finding it hard to prove this. Particularly, I am having trouble showing:
1) Take a line M in the plane and consider a point Q not on that line M. It should be the case that some other line, say M', contains Q and is parallel to M.
I intuitively get this, and I observed with the Fano Plane that removing one line causes certain lines to become parallel with one another. But, I still can't formulate a general argument to prove 1).
Let $L$ be the line you remove from the projective plane, the "line at infinity". Let $M$ be another line, and $Q$ a point not on $M$ and not on $L.$ Let $P$ be the point where $M$ intersects $L$ and let $M'$ be the line through $Q$ and $P.$ The lines $M$ and $M'$ are parallel since they meet at $P$ which is a point on $L.$