1. Tour of Fano’s Geometry
Axioms:
1. There exists at least one line.
2. Every line of the geometry has exactly three points on it.
3. Not all points of the geometry are on the same line.
4. For each two distinct points, there exists exactly one line on
both of them.
5. Each two lines have at least one point on both of them.

2. A Quick Theorem
Theorem (Theorem 1.7)
Each two distinct lines have exactly one point in common.
Proof.
Suppose 1 and 2 are distinct lines in Fano’s Geometry. Axiom 5
says there is at least one point on both of them. If points P and
P are on both 1 and 2 then Axiom 4 is contradicted as there are
two lines on both P and P .

14. What we’ve shown
Because every addition we made was strictly required to fulfill the
axioms, we’ve established two facts:
There are at least 7 points in Fano’s Geometry.
There are at least 7 lines in Fano’s Geometry.
In what seems to be the usual way, we can prove that there are no
more than 7 points in Fano’s Geometry by using an argument by
contradiction.

15. Part of a theorem
Theorem
Fano’s Geometry has exactly 7 points.
Proof.
We’ve established that there are at least 7 points. Suppose there is
an eighth, P8. By Axiom 4, there must be a line, that contains
both P1 and P8. Since must intersect the line that contains P3,
P7, and P4, and that line contains no other points, it follows that
contains either P3, P7, or P4. This contradicts Axiom 4.

16. The rest of the theorem
Theorem
Fano’s Geometry has exactly 7 lines.
Proof.
Suppose Fano’s geometry had a line that was not among the 7
we’ve established. By Axiom 4, each two distinct points must have
exactly one line on both of them, thus can contain at most one
of these 7 points. By axiom 2, though, contains 3 points.
Therefore, contains at least two points that are not among the
original 7. This contradicts our previous argument, that there are
exactly 7 points in the geometry.