  In the theory of hyperplane arrangements, the most important and difficult
problem is the combinatorial dependency of several properties. In this atricle,
we prove that Terao's celebrated addition-deletion theorem for free
arrangements is combinatorial, i.e., whether you can apply it depends only on
the intersection lattice of arrangements. The proof is based on a classical
technique. Since some parts are already completed recently, we prove the rest
part, i.e., the combinatoriality of the addition theorem. As a corollary, we
can define a new class of free arrangements called the additionally free
arrangement of hyperplanes, which can be constructed from the empty arrangement
by using only the addition theorem. Then we can show that Terao's conjecture is
true in this class. As an application, we can show that every ideal-Shi
arrangement is additionally free, implying that their freeness is
combinatorial.
