The attempted proof:
Given $P$ not on line $l$, line $PQ$ perpendicular to $l$ at $Q$, line $m$ perpendicular to $PQ$ at $P$ and point $A \neq P$ on $m$. Then, let $PB$ be the last ray between rays $PA$ and $PQ$ that intersects $l$, $B$ being the point of intersection. There exists a point $C$ on $l$ such that $Q * B * C$ (B is between $Q,C$). It follows that the ray $PB$ is not the last ray between rays $PA$ and $PQ$ that intersects $l$, and hence all rays between $PA$ and $PQ$ meet $l$. Thus $m$ is the only parallel to $l$ through $P$.
I can't seem to find it. I'm not sure if it is wrong because he is starting off with right angles, or something else. I feel that because he starts off with right angles, and this clearly only works for right angles, he is basically stating the parallel postulate for all the rays between the two lines as reasons for those not working. By stating and using the parallel postulate, he is being inconsistent. Am I on the right track?