  We prove the long standing conjecture in the theory of two-point boundary
value problems that completeness and Dunford's spectrality imply Birkhoff
regularity. In addition we establish the even order part of S.G.Krein's
conjecture that dissipative differential operators are Birkhoff-regular and
give sharp estimate of the norms of spectral projectors in the odd case.
  Considerations are based on a new direct method, exploiting \textit{almost
orthogonality} of Birkhoff's solutions of the equation $l(y)=\lambda y$, which
was discovered earlier by the author.
