  We present observations of [NII] 205 $\mu$m, [OIII] 88 $\mu$m and dust
emission in a strongly-lensed, submillimeter galaxy (SMG) at $z=6.0$,
G09.83808, with the Atacama Large Millimeter/submillimeter Array (ALMA). Both
[NII] and [OIII] line emissions are detected at $>12\sigma$ in the
0.8$"$-resolution maps. Lens modeling indicates that the spatial distribution
of the dust continuum emission is well characterized by a compact disk with an
effective radius of 0.64$\pm$0.02 kpc and a high infrared surface brightness of
$\Sigma_\mathrm{IR}=(1.8\pm0.3)\times10^{12}~L_\odot$ kpc$^{-2}$. This result
supports that G09.83808 is the progenitors of compact quiescent galaxies at
$z\sim4$, where the majority of its stars are expected to be formed through a
strong and short burst of star formation. G09.83808 and other lensed SMGs show
a decreasing trend of the [NII] line to infrared luminosity ratio with
increasing continuum flux density ratio between 63 $\mu$m and 158 $\mu$m, as
seen in local luminous infrared galaxies (LIRGs). The decreasing trend can be
reproduced by photoionization models with increasing ionization parameters.
Furthermore, by combining the [NII]/[OIII] luminosity ratio with far-infrared
continuum flux density ratio in G09.83808, we infer that the gas phase
metallicity is already $Z\approx 0.5-0.7~Z_\odot$. G09.83808 is likely one of
the earliest galaxies that has been chemically enriched at the end of
reionization.
