  An ATAI (or ATAF, respectively) algebra, introduced in [Jiang1] (or in [Fa]
respectively) is an inductive limit
$\lim\limits_{n\rightarrow\infty}(A_{n}=\bigoplus\limits_{i=1}A_{n}^{i},\phi_{nm})$,
where each $A_{n}^{i}$ is a simple separable nuclear TAI (or TAF) C*-algebra
with UCT property. In [Jiang1], the second author classified all ATAI algebras
by an invariant consisting orderd total K-theory and tracial state spaces of
cut down algebras under an extra restriction that all element in $K_{1}(A)$ are
torsion. In this paper, we remove this restriction, and obtained the
classification for all ATAI algebras with the Hausdorffized algebraic
$K_{1}$-group as an addition to the invariant used in [Jiang1]. The theorem is
proved by reducing the class to the classification theorem of $\mathcal{AHD}$
algebras with ideal property which is done in [GJL1]. Our theorem generalizes
the main theorem of [Fa] and [Jiang1] (see corollary 4.3).
