  In the paper the closed Friedmann--Robertson--Walker model with quantization
in the presence of the positive cosmological constant and radiation is studied.
For analysis of tunneling probability for birth of an asymptotically deSitter,
inflationary Universe as a function of the radiation energy a new definition of
a "free" wave propagating inside strong fields is proposed. On such a basis,
tunneling boundary condition is corrected, penetrability and reflection
concerning to the barrier are calculated in fully quantum stationary approach.
For the first time non-zero interference between the incident and reflected
waves has been taken into account which turns out to play important role inside
cosmological potentials and could be explained by non-locality of barriers in
quantum mechanics. Inside whole region of energy of radiation the tunneling
probability for the birth of the inflationary Universe is found to be close to
its value obtained in semiclassical approach. The reflection from the barrier
is determined for the first time (which is essentially differs on 1 at the
energy of radiation close to the barrier height). The proposed method could be
easily generalized on the cosmological models with the barriers of arbitrary
shape, that has been demonstrated for the FRW-model with included Chaplygin
gas. Result is stable for variations of the studied barriers, accuracy are
found to be 11--18 digits for all coefficients and energies below the barrier
height.
