The technical details of the new bootstrap method of Shimodaira (2004) are given here in mathematical proofs as well as a supporting computer program. Approximately unbiased tests based on the bootstrap probabilities are considered in Shimodaira (2004) for the exponential family of distributions with unknown expectation parameter vector, where the null hypothesis is represented as an arbitrary-shaped region with smooth boundaries. It has been described in the lemmas of Shimodaira (2004) that the newly developed three-step multiscale bootstrap method calculates an asymptotically third-order accurate $p$-value. All the mathematical proofs of these lemmas are shown here. The straightforward, though very tedious, calculations involving tensor notations are verified in Shimodaira (2003), which is, in fact, a computer program for Mathematica. Here we also give a brief explanation of this program.