  The boundary conditions on multiply connected extra dimensions play major
rolls in gauge-Higgs unification theory. Different boundary conditions, having
been given in ad hoc manner so far, lead to different theories. To solve this
arbitrariness problem of boundary conditions, we construct a formulation of
gauge-Higgs unification with dynamics of boundary conditions on M^4 times
S^1/Z_2. As a result, it is found that only highly restricted sets of boundary
conditions, which lead to nontrivial symmetry breaking, practically contribute
to the partition function. In particular, we show that for $SU(5)$ gauge group,
sets of boundary conditions which lead to SU(5) to SU(3) times SU(2) times U(1)
symmetry breaking are naturally included in the restricted sets.
