I need understand the proof of this Corollary given to Andrew Thomason,

Let $G$ be a multigraph, let $u, u \in V$, and suppose that $d ( w )$ is odd for each vertex $w \in V-$ { $u,v$ }. Then the number of Hamiltonian paths in $G$ from $u$ to $v$ is even.

I read this 'proof' but I don't understand the end.

I need understand the proof of this Corollary, attributed to Andrew Thomason.

Let $G$ be a multigraph, let $u, v \in V$, and suppose that $d ( w )$ is odd for each vertex $w \in V-$ { $u,v$ }. Then the number of Hamiltonian paths in $G$ from $u$ to $v$ is even.

I read this 'proof' but I don't understand the end.

I need understand the proof of this Corollary given to Andrew Thomason,

Let $G$ be a multigraph, let $u, u \in V$, and suppose that $d ( w )$ is odd for each vertex $w \in V-$ { $u,v$ }. Then the number of hamiltonian paths in $G$ from $u$ to $v$ is euen.

I read this 'proof'(Proofs of parity results via the Handshaking lemma) but I don´t understand the end.

I need understand the proof of this Corollary given to Andrew Thomason,

Let $G$ be a multigraph, let $u, u \in V$, and suppose that $d ( w )$ is odd for each vertex $w \in V-$ { $u,v$ }. Then the number of Hamiltonian paths in $G$ from $u$ to $v$ is even.

I read this 'proof' but I don't understand the end.