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We complete the four-dimensional ${\cal N}=1$ superfield description ofsix-dimensional supergravity. The missing ingredients in the previous works arethe superfields that contain the sechsbein $e_4^{\;\;\underline{\nu}}$,$e_5^{\;\;\underline{\nu}}$, $e_\mu^{\;\;\underline{4}}$,$e_\mu^{\;\;\underline{5}}$ and the second gravitino. They are necessary tomake the action invariant under the diffeomorphisms and the Lorentztransformations involving the extra dimensions. We find the correspondingsuperfield transformation laws, and show the invariance of the action underthem. We also check that the resultant action reproduces the known superfielddescription of five-dimensional supergravity through the dimensional reduction.

Full diffeomorphism and Lorentz invariance in 4D ${\cal N}=1$ superfield description of 6D SUGRA