Camion's theorem: every strongly connected tournament on at least three vertices has a directed Hamilton cycle

Strong connectivity gives a positive directed closed walk. Choose one of minimum positive length. It cannot have length one or two by [F1], and minimality prevents a repeated internal vertex, so it is a directed cycle of length at least three. By finiteness, choose a directed cycle $C$ of greatest length.

Assume for contradiction that $C$ omits at least one vertex. By [L1] and maximality, every vertex outside $C$ either sends arcs to every vertex of $C$ or receives arcs from every vertex of $C$. Call these two classes $A$ and $B$, respectively.

Strong connectivity forces both $A$ and $B$ to be nonempty: if $A$ were empty no outside vertex could reach $C$, while if $B$ were empty $C$ could not reach an outside vertex. A shortest directed path from $C$ to a vertex of $A$ begins by entering $B$ and at some first transition contains an arc $b a$ with $b\in B$ and $a\in A$.

Choose any cycle arc $c_i c_{i+1}$. Since $b\in B$ and $a\in A$, the arcs $c_i b$, $ba$, and $a c_{i+1}$ all occur. Replacing $c_i c_{i+1}$ by this three-arc path produces a directed cycle containing every vertex of $C$ together with $a,b$, contradicting the maximality of $C$.

The supposition in step 2.1 is impossible. Hence $C$ contains every vertex of $T$ and is a directed Hamilton cycle.