  TESS is revolutionising the search for planets orbiting bright and nearby
stars. In sectors 3 and 4, TESS observed TOI-402 (TIC-120896927), a bright
V=9.1 K1 dwarf also known as HD 15337, and found two transiting signals with
period of 4.76 and 17.18 days and radius of 1.90 and 2.21\,\Rearth. This star
was observed as part of the radial-velocity search for planets using the HARPS
spectrometer, and 85 precise radial-velocity measurements were obtained over a
period of 14 years. In this paper, we analyse the HARPS radial-velocity
measurements in hand to confirm the planetary nature of these two signals. By
reanalysing TESS photometry and host star parameters using EXOFASTv2, we find
that TOI-402.01 and TOI-402.02 have periods of 4.75642$\pm$0.00021 and
17.1784$\pm$0.0016 days and radii of 1.70$\pm$0.06 and
2.52$\pm$0.11\,\Rearth\,(precision 3.6 and 4.2\%), respectively. By analysing
the HARPS radial-velocity measurements, we find that those planets are both
super-Earths with masses of 7.20$\pm$0.81 and
8.79$\pm$1.67\,\Mearth\,(precision 11.3 and 19.0\%), and small eccentricities
compatible with zero at 2$\sigma$. Although having rather similar masses, the
radius of these two planets is really different, putting them on different
sides of the radius gap. With stellar irradiation 160 times more important than
Earth for TOI-402.01 and only 29 times more for TOI-402.02, it is likely that
photo-evaporation is at the origin of this radius difference. Those two
planets, being in the same system and therefore being in the same irradiation
environment are therefore extremely important to perform comparative
exoplanetology across the evaporation valley and thus bring constraints on the
mechanisms responsible for the radius gap.
