  A proof labelling scheme for a graph class $\mathcal{C}$ is an assignment of
certificates to the vertices of any graph in the class $\mathcal{C}$, such that
upon reading its certificate and the certificates of its neighbors, every
vertex from a graph $G\in \mathcal{C}$ accepts the instance, while if $G\not\in
\mathcal{C}$, for every possible assignment of certificates, at least one
vertex rejects the instance. It was proved recently that for any fixed surface
$\Sigma$, the class of graphs embeddable in $\Sigma$ has a proof labelling
scheme in which each vertex of an $n$-vertex graph receives a certificate of at
most $O(\log n)$ bits. The proof is quite long and intricate and heavily relies
on an earlier result for planar graphs. Here we give a very short proof for any
surface. The main idea is to encode a rotation system locally, together with a
spanning tree supporting the local computation of the genus via Euler's
formula.
