  Extending a result of Christiansen, we prove that every mutli-graph $G=(V,E)$
admits a proper edge colouring $\phi:E\to \{1,2,\dots\}$ which is local, that
is, $\phi(e)\le \max\{d(x)+\pi(x),d(y)+\pi(y)\}$ for every edge $e$ with
end-points $x,y\in V$, where $d(z)$ (resp.\ $\pi(z)$) denotes the degree of a
vertex $z$ (resp.\ the maximum edge multiplicity at $z$). This is derived from
a local version of the Fan Equation.
