We present new HST far-UV spectroscopy of two dozen hot evolved stars in NGC 2808, a massive globular cluster with a large population of "blue-hook" stars. The blue-hook stars are found in ultraviolet color-magnitude diagrams of the most massive globular clusters, where they fall at luminosities immediately below the hot end of the horizontal branch (HB), in a region of the HR diagram unexplained by canonical stellar evolution theory. Using new theoretical evolutionary and atmospheric models, we have shown that these subluminous HB stars are very likely the progeny of stars that undergo extensive internal mixing during a late He-core flash on the white dwarf cooling curve. This flash mixing leads to hotter temperatures and an enormous enhancement of the surface He and C abundances; the hotter temperatures and associated decrease in the hydrogen opacity shortward of the Lyman limit makes the stars brighter in the extreme UV but appear sub luminous in the UV and optical. Our far-UV spectroscopy demonstrates that, relative to normal HB stars at the same color, the blue-hook stars of NGC 2808 are hotter and greatly enhanced in He and C, thus providing unambiguous evidence of flash mixing in the subluminous population. Although the C abundance in the blue-hook stars is orders of magnitude larger than that in the normal HB stars, the atmospheric C abundance in both the blue-hook and normal HB stars appears to be affected by gravitational settling. The abundance variations seen in C, Si, and the Fe-peak elements indicate that atmospheric diffusion is at play in our sample, with all of our hot subdwarfs at 25,000 K to 50,000 K exhibiting large enhancements of the iron-peak elements. The hottest subdwarfs in our blue-hook sample may be pulsators, given that they fall in the temperature range of newly-discovered pulsating subdwarfs in omega Cen
