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Theory alternative to the vortex lattice melting theories is advertised. Thevortex lattice melting theories are science fiction cond-mat/9811051 becausethe Abrikosov state is not the vortex lattice with crystalline long-rangeorder. Since the fluctuation correction to the Abrikosov solution is infinitein the thermodynamic limit (K.Maki and H.Takayama, 1972) any fluctuation theoryof the mixed state should consider a superconductor with finite sizes. Suchnonperturbative theory for the easiest case of two-dimensional superconductorin the lowest Landau level approximation is presented in this work. Thethermodynamic averages of the spatial average order parameter and of theAbrikosov parameter $\beta_{a}$ are calculated. It is shown that the positionH_{c4} of the transition into the Abrikosov state (i.e. in the mixed state withlong-range phase coherence) depends strongly on sizes of two-dimensionalsuperconductor. Fluctuations eliminate the Abrikosov vortex state in a wideregion of the mixed state of thin films with real sizes and without pinningdisorders, i.e. H_{c4} << H_{c2}. The latter has experimental corroboration inPhys.Rev.Lett. 75, 2586 (1995).Comment: 4 pages, 0 figure

Existence of the Abrikosov vortex state in two-dimensional type-II superconductors without pinning