Fano's Geometry

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Axiom 1 There exists at least one line
Axiom 2 There are exactly 3 points on every line
Axiom 3 Not all points are on the same line
Axiom 4 There is exactly one line on any two distinct points.
Axiom 5 There is at least one point on any two distinct lines.

I've drawn a model and proven there are exactly seven lines and seven points. Each point is on exactly 3 lines. And, the set of all lines on any point contains all the points in the geometry.

Now I've been asked to prove that for any pair of points, there exists exactly 2 lines containing neither point.
I am stuck as to how to tackle this proof. Can someone get me started?
Thanks!
 
This proof really depends upon the fact that there exactly seven points.
Which follows from the first theorem: “Every two lines have exactly one point in common”.

For your proof, begin with two points A & B.
By axiom 4, there is a unique line k containing those two points.
There a point C distinct from A or B on k: axiom 2.
There is a point D not on k: axiom 3.
There is a line j containing C & D: axiom 4.
Can you prove that k is not j?
There a point E distinct from C or D on j: axiom 2.
There is a line m containing A & E: axiom 4.
Can you prove that m is not j or k?
There a point F distinct from A or E on m: axiom 2.

Now you use the line containing F & D to finish.
 
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