  We calculate the entanglement entropy of a non-contiguous subsystem of a
chain of free fermions. The starting point is a formula suggested by Jin and
Korepin, \texttt{arXiv:1104.1004}, for the reduced density of states of two
disjoint intervals with lattice sites $P=\{1,2,\dots,m\}\cup\{2m+1,2m+2,\dots,
3m\}$, which applies to this model. As a first step in the asymptotic analysis
of this system, we consider its simplification to two disjoint intervals
separated just by one site, and we rigorously calculate the mutual information
between these two blocks and the rest of the chain. In order to compute the
entropy we need to study the asymptotic behaviour of an inverse Toeplitz matrix
with Fisher-Hartwig symbol using the the Riemann--Hilbert method.
