(related to Proposition: Common Points of Two Distinct Straight Lines in a Plane)

- Let $g,h$ be two distinct straight lines in a plane $\alpha.$
- By 3rd axiom of connection there are at least two distinct points $A, B$ that lie on $g.$
- For sure, $A$ and $B$ do not lie both on $h$ since otherwise they would determine both straight lines $g$ and $h$ (2nd axiom of connection) and, by hypothesis, $g\neq h.$
- Therefore, at most one of the points $A$ and $B$ lie on both straight lines at once.∎

**Hilbert, David**: "Grundlagen der Geometrie", Leipzig, B.G. Teubner, 1903