  Theory alternative to the vortex lattice melting theories is advertised. The
vortex lattice melting theories are science fiction cond-mat/9811051 because
the Abrikosov state is not the vortex lattice with crystalline long-range
order. Since the fluctuation correction to the Abrikosov solution is infinite
in the thermodynamic limit (K.Maki and H.Takayama, 1972) any fluctuation theory
of the mixed state should consider a superconductor with finite sizes. Such
nonperturbative theory for the easiest case of two-dimensional superconductor
in the lowest Landau level approximation is presented in this work. The
thermodynamic averages of the spatial average order parameter and of the
Abrikosov parameter $\beta_{a}$ are calculated. It is shown that the position
H_{c4} of the transition into the Abrikosov state (i.e. in the mixed state with
long-range phase coherence) depends strongly on sizes of two-dimensional
superconductor. Fluctuations eliminate the Abrikosov vortex state in a wide
region of the mixed state of thin films with real sizes and without pinning
disorders, i.e. H_{c4} << H_{c2}. The latter has experimental corroboration in
Phys.Rev.Lett. 75, 2586 (1995).
