Let $P$, $Q$, $M$ be three fixed points not on the same line. Let $X$ be a point not on the lines $L_{PQ}$ or $L_{QM}$. Let $L$ be a line through $X$ which is not parallel to $L_{PQ}$ or $L_{QM}$, and also is not equal to $L{PX}$ or $L_{QX}$ or $L_{MX}$ (see the image below). Then $L$ intersects $L_{PQ}$ in a point $Y\ne P$, and also $L$ intersects $L_{PM}$ in a point $Z\ne M$. Also note that $Y\ne Z$ (and the proof continues...).
What I can't quite see is: why exactly is $Y\ne Z$? It is intuitive that that should be so, but how to justify it? Thanks.
You have $Z\in L_{PM}$ and $Y\in L_{PQ}$. Assuming $Y=Z$ you get $Y\in L_{PM}\cap L_{PQ}=\{P\}$ (because $P$, $Q$ and $M$ are not on the same line). But it is given that $Y\ne P$. Contradiction.