  We have estimated elemental abundances of the planetary nebula (PN) Hen2-436
in the Sagittarius (Sgr) spheroidal dwarf galaxy using ESO/VLT FORS2,
Magellan/MMIRS, and Spitzer/IRS spectra. We have detected candidates of [F II]
4790A, [Kr III] 6826A, and [P II] 7875A lines and successfully estimated the
abundances of these elements ([F/H]=+1.23, [Kr/H]=+0.26, [P/H]=+0.26) for the
first time. We present a relation between C, F, P, and Kr abundances among PNe
and C-rich stars. The detections of F and Kr support the idea that F and Kr
together with C are synthesized in the same layer and brought to the surface by
the third dredge-up. We have estimated the N^2+ and O^2+ abundances using
optical recombination lines (ORLs) and collisionally excited lines (CELs). The
discrepancy between the abundance derived from the O ORL and that derived from
the O CEL is >1 dex. To investigate the status of the central star of the PN,
nebula condition, and dust properties, we construct a theoretical SED model
with CLOUDY. By comparing the derived luminosity and temperature of the central
star with theoretical evolutionary tracks, we conclude that the initial mass of
the progenitor is likely to be ~1.5-2.0 Msun and the age is ~3000 yr after the
AGB phase. The observed elemental abundances can be explained by a theoretical
nucleosynthesis model with a star of initial mass 2.25 Msun, Z=0.008 and LMC
compositions. We have estimated the dust mass to be 2.9x10^-4 Msun (amorphous
carbon only) or 4.0x10^-4 Msun (amorphous carbon and PAH). Based on the
assumption that most of the observed dust is formed during the last two thermal
pulses and the dust-to-gas mass ratio is 5.58x10^-3, the dust mass-loss rate
and the total mass-loss rate are <3.1x10^-8 Msun / yr and <5.5x10^-6 Msun / yr,
respectively. Our estimated dust mass-loss rate is comparable to a Sgr dwarf
galaxy AGB star with similar metallicity and luminosity.
