Straight Line Perpendicular to Plane from Point is Unique

Proof

 * Euclid-XI-13.png

Suppose it were possible to set up two straight lines $AB$ and $AC$ perpendicular to the plane of reference and on the same side.

Let a plane be drawn through $BA$ and $AC$.

From :
 * let the common section be the straight line $DAE$.

Therefore the straight lines $AB$, $AC$ and $DAE$ are all in the same plane.

We have that $AC$ is perpendicular to the plane of reference.

So from :
 * $AC$ is perpendicular to all the straight lines which meet it and are in the plane of reference.

But $DAE$ meets $AC$ and is in the plane of reference.

Therefore $\angle CAE$ is a right angle.

For the same reason, $\angle BAE$ is a right angle.

Therefore $\angle BAE = \angle CAE$.

But both are in the same plane, which is impossible.

Hence the result.