(related to Proposition: Common Points of a Plane and a Straight Line Not in the Plane)
- By hypothesis, let $\alpha$ be a plane and $g$ be a straight line that is not in $\alpha.$
- If there is no point $A,$ that lies on both, $g$ and $\alpha$ then there is nothing left to be proven.
- Therefore, let $A$ lie on both, $g$ and $\alpha.$
- Assume, there is another point $B\neq A$ that also lies on both, $g$ and $\alpha.$
- By assumption and 6th axiom of connection, $g$ would be in $\alpha,$ contradicting the hypothesis.
- Therefore, there is at most one point $A,$ that lies on both.
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- Hilbert, David: "Grundlagen der Geometrie", Leipzig, B.G. Teubner, 1903