Different definitions of the phase PHI(gi)(r) of the gauage-invariant
order parameter in type-II superconductors re compared. It is shown th
at all phases satysfy the same equations and the boundary conditions.
In a mixed state the multi-valued part of PHI(gi)(r) makes a dominant
contribution to the correlation function of the order parameter and le
ads to an off-diagonal long-range order destruction by thermal fluctua
tions for three-dimensional superconductors. Possible consequences of
long-range order destruction are discussed for type-II superconductors
in magnetic field.