A Finite Geometry Exercise 1.9.
The art of doing mathematics consists in finding that special case which contains all the germs of generality.
—David Hilbert (1862–1943)

Exercise 1.9. Prove Fano's Theorem 4. Fano's geometry consists of exactly seven lines.

The proof is essentially the dual of the proof of Fano's Theorem 2 with some minor adjustments where:

    Fano's Theorem 2 implies the Dual of Axiom 1.
    Fano's Theorem 3 is the Dual of Axiom 2.
    Fano's Theorem's 2 and 3 imply the Dual of Axiom 3.
    Axioms 4 and 5 imply the Dual of Axiom 4.
    Axiom 4 implies the Dual of Axiom 5.

You may either prove that Fano's geometry satisfies the principle of duality or make the appropriate modifications to the proof of Fano's Theorem 2.