Common Section of Two Planes is Straight Line

Proof
Let $P$ and $Q$ be two distinct planes that cut one another.

Suppose $x, y, z$ are distinct points of their common section.

By Two Intersecting Straight Lines are in One Plane, the lines:


 * $L_1$, through $x$ and $y$
 * $L_2$, through $x$ and $z$

cannot intersect, for $L_1$ and $L_2$ are in both planes $P$ and $Q$.

However, since $x$ lies on both $L_1$ and $L_2$, it follows that $L_1 = L_2$.

Since $x, y$ and $z$ were arbitrary, it follows that the common section of $P$ and $Q$ is a straight line.