Proof
(related to Proposition: Plane Determined by two Crossing Straight Lines)
 By hypothesis, let $g,h$ be straight lines and let $A$ be the only point that lies on both of them.
 By 3rd axiom of connection, there are at least two points on $g$ and at least two points on $h$. Let $B$ be an additional point on $g$ and $C$ be an additional point on $h.$
 Let $\alpha$ be a plane, in which both, $g$ is located (i.e., by 6th axiom of connection, $A, B$ lie on $\alpha$) and $h$ is located (i.e., by 6th axiom of connection, $A, C$ lie on $\alpha$).
 By construction, $A, B, C$ lie on $\alpha.$
 BY 5th axiom of connection, the points $A, B, C$ completely determine $\alpha.$
 Therefore, there is exactly one plane $\alpha$ such that both straight lines $g$, $h$ are in that plane.
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References
Bibliography
 Hilbert, David: "Grundlagen der Geometrie", Leipzig, B.G. Teubner, 1903