Harmonic lasing is a perspective mode of operation of X-ray FEL user facilities that allows to provide brilliant beams of higher energy photons for user experiments. Another useful application of harmonic lasing is so called Harmonic Lasing Self-Seeded Free Electron Laser (HLSS FEL) that allows to improve spectral brightness of these facilities. In the past, harmonic lasing has been demonstrated in the FEL oscillators in infrared and visible wavelength ranges, but not in high-gain FELs and not at short wavelengths. In this paper we report on the first evidence of the harmonic lasing and the first operation of the HLSS FEL at the soft X-ray FEL user facility FLASH in the wavelength range between 4.5 nm and 15 nm. Spectral brightness was improved in comparison with Self-Amplified Spontaneous emission (SASE) FEL by a factor of six in the exponential gain regime. A better performance of HLSS FEL with respect to SASE FEL in the post-saturation regime with a tapered undulator was observed as well. The first demonstration of harmonic lasing in a high-gain FEL and at short wavelengths paves the way for a variety of applications of this new operation mode in X-ray FELs.