We present observations of [N II] 205 mu m, [O III]88 mu m, and dust emission in a stronglylensed, submillimeter galaxy (SMG) at z= 6.0, G09.83808, with the Atacama Large Millimeter/submillimeter Array (ALMA). Both [N II] and [O III] 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 +/- 0.02 kpc and a high infrared surface brightness of Sigma(IR) = (1.8 +/- 0.3) x 10(12) L-circle dot kpc(-2). This result supports that G09.83808 is the progenitor of compact quiescent galaxies at z similar to 4, 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 in the [N Id line to infrared luminosity ratio with increasing continuum flux density ratio between 63 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 [N II]/[O III] luminosity ratio with far-infrared continuum flux density ratio in G09.83808, we infer that the gas phase metallicity is already Z approximate to 0.5-0.7 Z(circle dot). G09.83808 is likely one of the earliest galaxies that has been chemically enriched at the end of reionization.