Uniqueness theorem for static dilaton $\mathrm{U}{\mathrm{}\left(1\right)\mathrm{}}^2$ black holes

Using a sigma model formulation of the field equations as on a two-dimensional manifold we provide the proof of a black hole uniqueness solution in $N=4,d=4\mathrm{}$ supergravity subject to certain boundary conditions. We considered the black hole solutions both in $\mathrm{SU}\left(4\right)\mathrm{}$ and $\mathrm{SO}\left(4\right)\mathrm{}$ versions of the underlying theory.