Straight Line which cuts Harmonic Pencil forms Harmonic Range

Theorem
Let $AB$ and $PQ$ be line segments on a straight line such that $\tuple {AB, PQ}$ is a harmonic range.

Let $O$ be a point which is not on the straight line $AB$.

Let $\map O {AB, PQ}$ be the harmonic pencil formed from $O$ and $\tuple {AB, PQ}$.

Let a straight line intersect $OA$, $OB$, $OP$ and $OQ$ at $A'$, $B'$, $P'$ and $Q'$ respectively.


 * Line-intersecting-harmonic-pencil.png

Then $A'$, $B'$, $P'$ and $Q'$ themselves form a harmonic range $\tuple {A'B', P'Q'}$.