  We treat the lattice sine-Gordon equation and two of its generalised
symmetries as a compatible system. Elimination of shifts from the two
symmetries of the lattice sine-Gordon equation yields an integrable NLS-type
system. An auto-B\"acklund transformation and a superposition formula for the
NLS-type system is obtained by elimination of shifts from the lattice
sine-Gordon equation and its down-shifted version. We use the obtained formulae
to calculate a superposition of two and three elementary solutions.
