The ring Z_(2) consists of rational numbers that can be written with an odd denominator, equivalently those with nonnegative two-adic valuation. It contains 1/3 but not 1/2. Reduction modulo 2^p means multiplying the numerator residue by the inverse of its odd denominator, not rounding the rational number. For example, 1/3 is 3 modulo 8 because 3 times 3 is 1 modulo 8. In this proof, integral often means belonging to this local ring or its extension, not being an ordinary integer.