We present a general formulation of chiral gauge theories, which admits Dirac operators with more general spectra, reveals considerably more possibilities for the structure of the chiral projections, and nevertheless allows appropriate realizations. In our analyses we use two forms of the correlation functions which both also apply in the presence of zero modes and for any value of the index. To account properly for the conditions on the bases the concept of equivalence classes of pairs of them is introduced. The behaviors under gauge transformations and under CP transformations are unambiguously derived.
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