- 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.

q.e.d

| | | | created: 2019-12-26 12:15:28 | modified: 2019-12-26 12:15:44 | by: *bookofproofs* | references: [8324]

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[8324] **Hilbert, David**: “Grundlagen der Geometrie”, Leipzig, B.G. Teubner, 1903