  We study the Higgs boson search possibility at the TEVATRON energies. The
detailed analysis of the 4-fermion final state reactions is presented. Among
the complete set of Feynman diagrams the main contribution to the final states
comes from the Higgs signal WH and ZH, from the backgrounds WZ, ZZ and gluonic
background Wbb, Zbb with subssiquent decays of W, Z bosons. It is shown that
the important contribution to the background is connected with single top quark
production subreactions qq-bar --> tb-bar or t-bar b with the top decay to Wb.
As a result of various distributions for Higgs signal and backgrounds a number
of optimum cuts has been found. The finaleffective $b\bar{b}$ -- pair mass
distribution allows to extract the signal from Higgs with mass up to about 100
GeV at 2 TeV C.M. energy and up to about 120 GeV at 4 TeV C.M. energy for the
integrated luminosity 1000 inverse pb.
